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determinant

Y ...a n v, the summation being for all permutations of the n numbers, is called the determinant of the n 2 quantities.

are called the elements of the determinant; the term (-) k alaa20a37...anv is called a member of the determinant, and there are evidently n!

The determinant is usually written all a12 a13.

Each row as well as each column supplies one and only one element to each member of the determinant.

Consideration of the definition of the determinant shows that the value is unaltered when the suffixes in each element are transposed.

If the determinant is transformed so as to read by columns as it formerly did by rows its value is unchanged.

The leading member of the determinant is alla22a33ï¿½ï¿½ï¿½ann, and corresponds to the principal diagonal of the matrix.

We write frequently 0 = alla22a33ï¿½ï¿½ï¿½ann = (ana22a33ï¿½ï¿½ï¿½ann)ï¿½ If the first two columns of the determinant be transposed the ' The elementary theory is given in the article Determinant.

expression for the determinant becomes Z(-) k aitia2aa3y...anv, viz.

Hence the transposition of columns merely changes the sign of the determinant.

Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant.

Interchange of any two rows or of any two columns merely changes the sign of the determinant.

If any two rows or any two columns of a determinant be identical the value of the determinant is zero.

From the value of A we may separate those members which contain a particular element a ik as a factor, and write the portion aik A ik; A k, the cofactor of ar k, is called a minor of order n - i of the determinant.

Hence anAu = auk t a22a33...ann, where the cofactor of an is clearly the determinant obtained by erasing the first row and the first column.

a ll a33 ï¿½ï¿½ï¿½ a32 a33 ï¿½ï¿½ï¿½ a3n an2 an3 ï¿½ï¿½ï¿½ ann Similarly A ik, the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column.

No member of a determinant can involve more than one element from the first row.

Since the determinant having two identical rows, and an3 an3 ï¿½ï¿½ï¿½ ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+ï¿½ï¿½ï¿½ +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+ï¿½ï¿½ ï¿½ +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+ï¿½ï¿½ï¿½ +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+ï¿½ï¿½ï¿½+aniAnk = 0 (k .>

Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets.

The minor Aik is aa, and is itself a determinant of order n-t.

In obtaining the minor Aik in the form of a determinant we erased certain rows and columns, and we would have erased in an exactly similar manner had we been forming the determinant associated with A2:8, since the deleting lines intersect rk in two pairs of points.

In the latter case the sign is determined by -I raised to the same power as before, with the exception that Tux., replaces Tusk; but if one of these numbers be even the other must be uneven; hence A ik = - Aisï¿½ rk Moreover aik a,, aikarsAik +aisarkAis Aik, rk aik ars rs where the determinant factor is giyen by the four points in which the deleting lines intersect.

This determinant and that associated with Aik are termed corresponding determinants.

From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ï¿½ï¿½ï¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.

Now by the expansion theorem the determinant becomes (-)1 +2+3+ï¿½.ï¿½+2nB.0 = (- I)n(2n +1) +nC =C.

We thus obtain for the product a determinant of order n.

We may say that, in the resulting determinant, the element in the ith row and k th column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklb11+ak2b12+ak3b13ï¿½ï¿½ï¿½ +aknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.

In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.

The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 ï¿½ï¿½ï¿½ A nn).

Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero.

Its value is therefore O n and we have the identity D.0 = A n or D It can now be proved that the first minor of the adjoint determinant, say B rs is equal to An-2aï¿½.

The adjoint determinant is the (n - I) th power of the original determinant.

The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by aik = aki.

It is easy to see that the adjoint determinant is also 'symmetrical, viz.

such that Aik=Aki, for the determinant got by suppressing the ith row and k th column differs only by an interchange of rows and columns from that got by suppressing the k th row and i th column.

Now the determinant has the value - {AiA11+A2A22+A3A33+2A2A3A23+2A3AIA31+2A1A2Al2{ = -Eata r r-2EA r A 8 A rs in general, and hence by substitution {A I V A n+ A 211 A22+ï¿½ï¿½ï¿½ +A71 Ann}2.

A skew symmetric determinant has a,.

When a skew symmetric determinant is of even degree it is a perfect square.

In the case of the determinant of order 4 the square root is Al2A34 - A 13 A 24 +A14A23.

A skew determinant is one which is skew symmetric in all respects,.

Such a determinant is of importance in the theory of orthogonal substitution.

Let the determinant of the b's be Ab and B rs, the minor corresponding to b rs .

We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.

determinant we derive and thence 'dx' n ay2 ?ay?

x n / I l yl, y2,ï¿½ï¿½ï¿½yn Theorem.-If the functions y 1, y2,ï¿½ï¿½ï¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.

Hence_li y ` A 1n where A li and A li, are minors of the complete determinant (a11a22...ann)ï¿½ an1 ant ï¿½ï¿½ï¿½an,n-1 or, in words, y i is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the n th column, multiplied by (-r)i+n.

Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.

+al pxp = 0, a21x1 +a22x2 + ï¿½ ï¿½ ï¿½ +a2pxp = 0, aplxl+ap2x2+...+appxp = 0, be the system the condition is, in determinant form, (alla22...app) = 0; in fact the determinant is the resultant of the equations.

Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n.

He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m.

We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant.

For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=ï¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

The expression in form of a determinant presents in general considerable difficulties.

If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.

Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.

= (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

The identities are, in particular, of service in reducing symbolic products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor (ab) is even.

To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.

Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...

Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, ï¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2ï¿½ It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.

The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants.

A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

We have A +k 1 f =0 2, O+k 2 f = x2, O+k3f =4) 2, and Cayley shows that a root of the quartic can be xpressed in the determinant form 1, k, 0.1y the remaining roots being obtained by varying 1, k, x the signs which occur in the radicals 2 u The transformation to the normal form reduces 1, k 3, ?

Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance.

If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.

The number of different symbols a, b, c,...denotes the the covariants are homogeneous, but not in general isobaric functions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities: - (ï¿½b) 2 2)2 = a b - a b; (xï¿½ = as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors.

if the determinant factors to be, each of them, either zero or unity.

Or, if we please, we may leave the determinant factors untouched and consider the exponents ji, j2, j3,ï¿½ï¿½ï¿½11, 12, 1 3, ...

"If the sovereign power is to be understood in this fuller, less abstract sense, if we mean by it the real determinant of the habitual obedience of the people, we must look for its sources much more widely and deeply than the analytical jurists do; it can no longer be said to reside in a determinate person or persons, but in that impalpable congeries of the hopes and fears of a people bound together by common interest and sympathy, which we call the common will" (Green's Works, 2.404).

But an attribute, though real, is not a distinct reality, but only a determinant of a substance, and has no being of its own apart from the substance so determined; whereas a substance, determined by all its attributes, is different from everything else in the world.

Colour, therefore, must be correlated with some determinant (determining factor) for pattern, and it cannot, therefore, exist alone in an animal's coat.

And we must conceive that each kind of pattern - the self, the spotted, the striped, the hooded and all others - has its own special determinant.

When an albino mouse, rat, guinea-pig or rabbit is crossed with either a pure self or pure pied-coloured form, the offspring are similar to, though not always exactly like, the coloured parent; provided, of course, that the albino is pure and is not carrying some colour or pattern determinant which is dominant to that of the coloured parent used.

This conflict arises not only from naturalization having been granted without the corresponding expatriation having been permitted, but also from the fact that birth on the soil was the leading determinant of nationality by feudal law, and still is so by the laws of England and the United States (jus soli), while the nationality of the father is its leading determinant in those countries which have accepted Roman principles of jurisprudence (jus sanguinis).

The physiography of the state is the evident determinant of its climate, fauna and flora.

a~ are the minors of the rth row of the determinant (7).

determinant a1 - I a2 O -I o 0 u an bn O O - - o o -I a n, from which point of view continuants have been treated by W.

The idea, inasmuch as it is a law of universal mind, which in particular minds produces aggregates of sensations called things, is a "determinant" (iripas ixov), and as such is styled "quantity" and perhaps "number" but the ideal numbers are distinct from arithmetical numbers.

Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).

xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = o with the Hessian, or curve A = o, where A is the determinant formed with the second derived functions of u.

We might infer from this that the intellect, so judging, is itself the proper and complete determinant of the will, and that man, as a rational being, ought to aim at the realization of absolute good for its own sake.

With Price, again, he holds that rightness of intention and motive is not only an indispensable condition or element of the rightness of an action, but actually the sole determinant of its moral worth; but with more philosophical consistency he draws the inference - of which the English moralist does not seem to have dreamt - that there can be no separate rational principles for determining the " material " rightness of conduct, as distinct from its " formal " rightness; and therefore that all rules of duty, so far as universally binding, must admit of being exhibited as applications of the one general principle that duty ought to be done for duty's sake.

DETERMINANT, in mathematics, a function which presents itself in the solution of a system of simple equations.

Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

It is to be noticed that the resulting equation is a,b,c x= d,b,c,, ,, a' b' c' d'b' c' an, b", cn d", b", c" where the expression on the right-hand side is the like function with d, d', d" in place of a, a', a" respectively, and is of course also a determinant.

we have = a, 'l' a, b a', b'I a, b, c =ab', c' -Farb" a' b', c' b", c" b, c a", b" cn a, b, c, d a', b', c', d' a", b", c", d" a'" b in c'" d,n,, so on, the terms being all + for a determinant of an odd ord er, but alternately + and - for a determinant of an even order.

It is easy, by induction, to arrive at the general results: A determinant of the order n is the sum of the 1.2.3...n products which can be formed with n elements out of n 2 elements arranged in the form of a square, no two of the n elements being in the same line or in the same column, and each such product having the coefficient = unity.

The products in question may be obtained by permuting in every possible manner the columns (or the lines) of the determinant, and then taking for the factors the n elements in the dexter diagonal.

'This implies the theorem that a given arrangement can be derived from the primitive arrangement only by an odd number, or else only by an even number of interchanges, - a of which may be easily obtained from the theorem (in fact a particular case of the general one), an arrangement can be derived from itself only by an even number of interchanges.] And this being so, each product has the sign belonging to the corresponding arrangement of the columns; in particular, a determinant contains with the sign + the product of the elements in its dexter diagonal.

Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is =ab'c" - ab "c'+a'b "c - a'bc" - a"bc' - a"b'c. 3.

It further appears that a determinant is a linear function' of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs.

It at once follows that, if two columns are identical, or if two lines are identical, the value of the determinant is = o.

It may be added, that if the lines are converted into columns, and the columns into lines, in such a way as to leave the dexter diagonal unaltered, the value of the determinant is unaltered; the determinant is in this case said to be transposed.

By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.

Observe that the properties show at once that if any column is = o (that is, if the elements in the column are each = o), then the determinant is = o; and further, that if any two columns are identical, then the determinant is = o.

Reverting to the system of linear equations written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.

the second and third terms each vanishing, it is a, b, c a', c', , a", bn c",, But if the linear equations hold good, then the first column of the 1 The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is that the determinant is in regard to the elements a, a', a", ..

=x d,b,c d ', c', , d "b" c",, original determinant is = o, and therefore the determinant itself is = o; that is, the linear equations give x'a,b,c - d,b,c =o; a', b', c' d', b', c' a", b', c" d", b", c" which is the result obtained above.

- The theorem is obtained very easily from the last preceding definition of a determinant.

It is most simply expressed thus where the expression on the left side stands for a determinant, the terms"of the first line being (a, b, c) (a, a', a"), that is, as+ ba'+ ca", (a, b, c) (/3, /3', 13"), that is, a/3+b/3'+0", (a, b, c) (y, y, 'Y'), that is ay+by'+cy"; and similarly the terms in the second and third lines are the life functions with (a', b', c') and (a", b",c") respectively.

There is an apparently arbitrary transposition of lines and columns; the result would hold good if on the left-hand side we had written (a, (3, y), (a', 13', y'), (a", (3", y"), or what is the same thing, if on the right-hand side we had transposed the second determinant; and either of these changes would, it might be thought, increase the elegance of the form, but, for a reason which need not be explained,' the form actually adopted is the preferable one.

To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

broken up into a sum of (3 3 =) 2 7 determinants, each of which is either of some such form as a, a, a' a' b' a", a", where the term a/3y' is not a term of the a/3y-determinant, and its coefficient(as a determinant with two identical columns) vanishes; or else it is of a form such as t af'y".

a, b, c a' b' c', a"b" c", b", that is, every term which does not vanish contains as a factor the abc-determinant last written down; the sum of all other factors a0'y" is the a/37-determinant of the formula; and the final result then is, that the determinant on the left-hand side is equal to the product on the right-hand side of the formula.

- Consider, for simplicity, a determinant of the fifth order, 5= 2+3, and let the top two lines be a, b, c, d, e a', c' d'e', ,, then, if we consider how these elements enter into the determinant, it is at once seen that they enter only through the determinants of the second order a,: b' I, &c., which can be formed by selecting any two columns at pleasure.

Moreover, representing the remaining three lines by a" b" c" d" e" b /r c a, d N, e"' a " c 'N d"N err" it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the complementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form ' a b c" d" e" a, b"c"'dN, ear the sign being in each case such that the sign of the term .c"d"'e" obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously +.

Observe that for a determinant of the n-th order, taking the decomposition to be r + (n - I), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.

Any determinant I a,' b, I formed out of the elements of the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n - I, then such determinant is called a first minor; the number of the first minors is = n 2, the first minors, in fact, corresponding to the several elements of the determinant - that is, the coefficient therein of any term whatever is the corresponding first minor.

The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.

A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be = o), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is skew; but if the relation does extend to the diagonal terms (that is, if these are each = o), then the determinant is skew symmetrical; thus the determinants a, h, g a, v, - µ 0, v, - h, b, f - v, h, - v, 0, g,f,c c 12, - X, o are respectively symmetrical, skew and skew symmetrical: =0; a,b,c,d a' b' c' d'a" b c d" a, b, c, d a' b' c' d'a", b N' c N' dN,, , c d The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics.

Laplace developed a theorem of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.

To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant.

behaviour is a major determinant of smoking behavior.

determinant of the pathogenicity of many bacterial strains is the ability to resist complement-mediated destruction.

determinant of a square diagonal matrix is the product of its diagonal elements.

determinant of competitiveness.

The definition of a determinant in all dimensions will be given in detail, together with applications and techniques for calculating determinants.

They therefore represent a finite scientific and economic resource and are a notable determinant of landscape character.

The determinant of a square diagonal matrix is the product of its diagonal matrix is the product of its diagonal elements.

Here det must be an integer which is a multiple of the biggest determinant divisor of A.

The fermion determinant is calculated by summing over the resulting complex eigenvalues.

notable determinant of landscape character.

Key findings: Territorial behavior is a conspicuous determinant of social organization in many reef fishes including parrotfish.

The determinant of a permutation matrix equals the signature of the column permutation matrix equals the signature of the column permutation.

The determinant of a permutation matrix equals the signature of the column permutation.

subject matter of the text is, obviously, a crucial determinant of the role that spatial inferences play in understanding it.

Y ...a n v, the summation being for all permutations of the n numbers, is called the determinant of the n 2 quantities.

are called the elements of the determinant; the term (-) k alaa20a37...anv is called a member of the determinant, and there are evidently n!

The determinant is usually written all a12 a13.

Ã¯¿½Ã¯¿½ aln a a2n and ant an3 Ã¯¿½Ã¯¿½Ã¯¿½ ann the square array being termed the matrix of the determinant.

A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant.

Each row as well as each column supplies one and only one element to each member of the determinant.

Consideration of the definition of the determinant shows that the value is unaltered when the suffixes in each element are transposed.

If the determinant is transformed so as to read by columns as it formerly did by rows its value is unchanged.

The leading member of the determinant is alla22a33Ã¯¿½Ã¯¿½Ã¯¿½ann, and corresponds to the principal diagonal of the matrix.

We write frequently 0 = alla22a33Ã¯¿½Ã¯¿½Ã¯¿½ann = (ana22a33Ã¯¿½Ã¯¿½Ã¯¿½ann)Ã¯¿½ If the first two columns of the determinant be transposed the ' The elementary theory is given in the article Determinant.

expression for the determinant becomes Z(-) k aitia2aa3y...anv, viz.

Hence the transposition of columns merely changes the sign of the determinant.

Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant.

Interchange of any two rows or of any two columns merely changes the sign of the determinant.

If any two rows or any two columns of a determinant be identical the value of the determinant is zero.

From the value of A we may separate those members which contain a particular element a ik as a factor, and write the portion aik A ik; A k, the cofactor of ar k, is called a minor of order n - i of the determinant.

Hence anAu = auk t a22a33...ann, where the cofactor of an is clearly the determinant obtained by erasing the first row and the first column.

a ll a33 Ã¯¿½Ã¯¿½Ã¯¿½ a32 a33 Ã¯¿½Ã¯¿½Ã¯¿½ a3n an2 an3 Ã¯¿½Ã¯¿½Ã¯¿½ ann Similarly A ik, the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column.

No member of a determinant can involve more than one element from the first row.

A =ailAii+a12A12+a13A13+Ã¯¿½Ã¯¿½Ã¯¿½ +ainAin) (A) Ã¯¿½ A = lk + a2kA2k +a3kA3k +Ã¯¿½Ã¯¿½Ã¯¿½ +ankAnk This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved.

Since the determinant having two identical rows, and an3 an3 Ã¯¿½Ã¯¿½Ã¯¿½ ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+Ã¯¿½Ã¯¿½Ã¯¿½ +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+Ã¯¿½Ã¯¿½ Ã¯¿½ +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+Ã¯¿½Ã¯¿½Ã¯¿½ +aknAin =0 (k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+Ã¯¿½Ã¯¿½Ã¯¿½+aniAnk = 0 (k .>

If we multiply the elements of the second row by an arbitrary magnitude X, and add to the corresponding elements of the first row, A becomes Zai,A18+XEa28A13 = Lia13A18 =A, showing that the value of the determinant is unchanged.

- The value of a determinant is unchanged if we add to the elements of any row or column the corresponding elements of the other rows or other columns respectively each multiplied by an arbitrary magnitude, such magnitude remaining constant in respect of the elements in a particular row or a particular column.

Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets.

The minor Aik is aa, and is itself a determinant of order n-t.

In obtaining the minor Aik in the form of a determinant we erased certain rows and columns, and we would have erased in an exactly similar manner had we been forming the determinant associated with A2:8, since the deleting lines intersect rk in two pairs of points.

In the latter case the sign is determined by -I raised to the same power as before, with the exception that Tux., replaces Tusk; but if one of these numbers be even the other must be uneven; hence A ik = - AisÃ¯¿½ rk Moreover aik a,, aikarsAik +aisarkAis Aik, rk aik ars rs where the determinant factor is giyen by the four points in which the deleting lines intersect.

This determinant and that associated with Aik are termed corresponding determinants.

Since A lk is a determinant we similarly obtain Alk = a21Alk+Ã¯¿½ Ã¯¿½ Ã¯¿½ +a2,k-iAl,k +a2,k+lAl,k+ Ã¯¿½Ã¯¿½Ã¯¿½+a2 21 2,k-1 2, k +1 2,n and thence = Xalia2kAli where k; i,k 2k and as before A = a1, an A i> k i,k I ail, auk 12k an important expansion of A.

If the jth column be identical with the i ll ' the determinant A vanishes identically; hence if j be not equal to i, k, or r, a 11 a 21 a31 0 =I alk a2k a3k A11.

From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, Ã¯¿½Ã¯¿½Ã¯¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.

Now by the expansion theorem the determinant becomes (-)1 +2+3+Ã¯¿½.Ã¯¿½+2nB.0 = (- I)n(2n +1) +nC =C.

We thus obtain for the product a determinant of order n.

We may say that, in the resulting determinant, the element in the ith row and k th column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklb11+ak2b12+ak3b13Ã¯¿½Ã¯¿½Ã¯¿½ +aknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.

In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.

The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 Ã¯¿½Ã¯¿½Ã¯¿½ A nn).

Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero.

Its value is therefore O n and we have the identity D.0 = A n or D It can now be proved that the first minor of the adjoint determinant, say B rs is equal to An-2aÃ¯¿½.

From the equations a11xi+ a12x2+ a13x3 +Ã¯¿½Ã¯¿½Ã¯¿½ = El, a21x1+a72x2+ a23x3+Ã¯¿½Ã¯¿½Ã¯¿½ = 2, a3lxl+a32x2+a33x3+Ã¯¿½Ã¯¿½Ã¯¿½ = 53, 0x1 =A111+A21E2+A31Er3+Ã¯¿½Ã¯¿½Ã¯¿½ 0x2 = Al2E1 + A22E2+ A32Srr3+Ã¯¿½Ã¯¿½Ã¯¿½ AX3 =A13E1+A23E2+A33E3+Ã¯¿½Ã¯¿½Ã¯¿½ A n 1 E1 = B110x1 + B12Ax2+ B13Ax3+Ã¯¿½Ã¯¿½Ã¯¿½, On - lt2 = B 21Ax1+ B220x2+ B230x3+Ã¯¿½Ã¯¿½Ã¯¿½ An-15513 = B31Ax1 + B 32Ax2+B330x3+Ã¯¿½Ã¯¿½Ã¯¿½ and comparison of the first and third systems yields B = An-2a rs = rsÃ¯¿½ In general it can be proved that any minor of order of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the h power of the original determinant.

The adjoint determinant is the (n - I) th power of the original determinant.

The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by aik = aki.

It is easy to see that the adjoint determinant is also 'symmetrical, viz.

such that Aik=Aki, for the determinant got by suppressing the ith row and k th column differs only by an interchange of rows and columns from that got by suppressing the k th row and i th column.

If any symmetrical determinant vanish and be bordered as shown below all a12 a13 Al a12 a22 a23 A2 a13 a23 a33 A3 Al A2 A3 Ã¯¿½ it is a perfect square when considered as a function of A 11 A2, A3.

Now the determinant has the value - {AiA11+A2A22+A3A33+2A2A3A23+2A3AIA31+2A1A2Al2{ = -Eata r r-2EA r A 8 A rs in general, and hence by substitution {A I V A n+ A 211 A22+Ã¯¿½Ã¯¿½Ã¯¿½ +A71 Ann}2.

A skew symmetric determinant has a,.

Such a determinant when of uneven degree vanishes, for if we multiply each row by - I we multiply the determinant by (- I) n = -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered.

When a skew symmetric determinant is of even degree it is a perfect square.

In the case of the determinant of order 4 the square root is Al2A34 - A 13 A 24 +A14A23.

A skew determinant is one which is skew symmetric in all respects,.

Such a determinant is of importance in the theory of orthogonal substitution.

Let the determinant of the b's be Ab and B rs, the minor corresponding to b rs .

We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.

determinant we derive and thence 'dx' n ay2 ?ay?

x n / I l yl, y2,Ã¯¿½Ã¯¿½Ã¯¿½yn Theorem.-If the functions y 1, y2,Ã¯¿½Ã¯¿½Ã¯¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.

Denote by A the determinant (a11a22Ã¯¿½Ã¯¿½Ã¯¿½ann)Ã¯¿½ Multiplying the equations by the minors A l, .., A2,,,,Ã¯¿½Ã¯¿½Ã¯¿½Ani., respectively, and adding, we obtain x 1 (ai, Aig+a2p.A2lc+Ã¯¿½Ã¯¿½Ã¯¿½+anÃ¯¿½AnÃ¯¿½) =xÃ¯¿½A=o, since from results already given the remaining coefficients of x 11' x 2, ...x Ã¯¿½ 'i xÃ¯¿½+I,...x, vanish identically.

Hence_li y ` A 1n where A li and A li, are minors of the complete determinant (a11a22...ann)Ã¯¿½ an1 ant Ã¯¿½Ã¯¿½Ã¯¿½an,n-1 or, in words, y i is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the n th column, multiplied by (-r)i+n.

Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.

+al pxp = 0, a21x1 +a22x2 + Ã¯¿½ Ã¯¿½ Ã¯¿½ +a2pxp = 0, aplxl+ap2x2+...+appxp = 0, be the system the condition is, in determinant form, (alla22...app) = 0; in fact the determinant is the resultant of the equations.

Taking the same example as before the process leads to the system of equations acx 4 +alx 3 +a2x 2 +a3x =0, aox 3 +a1x 2 +a2x+a 3 = 0, box +bix -1-b2x =0, box' +b i x 2 -{-h 2 x = 0, box + b i x + b:: = 0, whence by elimination the resultant a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 a3 bo b 1 b 2 0 0 0 bo b 1 b 2000 bo b 1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former.

Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n.

He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m.

Put (aox 3 -}-a l x 2 +a 2 x +a 3) (box' +b1x'+b2) - (aox'3+aix'2+a2x'+a3) (box' + bix + b2) = 0; after division by x-x the three equations are formed aobcx 2 = aobix+aob2 =0, aobix 2 + (aob2+a1b1-a2bo) x +alb2 -a3bo = 0, aob2x 2 +(a02-a3bo)x+a2b2-a3b1 =0 and thence the resultant aobo ao aob2 aob 1 aob2+a1b1-a2bo alb2-a3b0 aob 2 a1b2 - a 3 bo a2b2 - a3b1 which is a symmetrical determinant.

We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant.

For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=Ã¯¿½.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J.

The expression in form of a determinant presents in general considerable difficulties.

If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that r must differ from zero.

Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.

= (A11+A22)n by the substitutions 51 = A l, E1+Ã¯¿½1 2, 52 = A2E1+Ã¯¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = Ã¯¿½la1 +Ã¯¿½2a2Ã¯¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

The identities are, in particular, of service in reducing symbolic products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor (ab) is even.

To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically.

Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...

Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, Ã¯¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1.

It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2Ã¯¿½ It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.

The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants.

A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form.

We have A +k 1 f =0 2, O+k 2 f = x2, O+k3f =4) 2, and Cayley shows that a root of the quartic can be xpressed in the determinant form 1, k, 0.1y the remaining roots being obtained by varying 1, k, x the signs which occur in the radicals 2 u The transformation to the normal form reduces 1, k 3, ?

Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance.

determinant factors (abc), (abd), (bce), etc...., and other factors az, bx, cx,...

If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.

The number of different symbols a, b, c,...denotes the the covariants are homogeneous, but not in general isobaric functions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities: - (Ã¯¿½b) 2 2)2 = a b - a b; (xÃ¯¿½ = as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors.

if the determinant factors to be, each of them, either zero or unity.

Or, if we please, we may leave the determinant factors untouched and consider the exponents ji, j2, j3,Ã¯¿½Ã¯¿½Ã¯¿½11, 12, 1 3, ...

He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see Algebraic Forms).

"If the sovereign power is to be understood in this fuller, less abstract sense, if we mean by it the real determinant of the habitual obedience of the people, we must look for its sources much more widely and deeply than the analytical jurists do; it can no longer be said to reside in a determinate person or persons, but in that impalpable congeries of the hopes and fears of a people bound together by common interest and sympathy, which we call the common will" (Green's Works, 2.404).

But an attribute, though real, is not a distinct reality, but only a determinant of a substance, and has no being of its own apart from the substance so determined; whereas a substance, determined by all its attributes, is different from everything else in the world.

Colour, therefore, must be correlated with some determinant (determining factor) for pattern, and it cannot, therefore, exist alone in an animal's coat.

And we must conceive that each kind of pattern - the self, the spotted, the striped, the hooded and all others - has its own special determinant.

They carry only some determinant or determinants which are capable of developing colour when they interact with some other determinant or determinants carried alone by pigmented individuals.

When an albino mouse, rat, guinea-pig or rabbit is crossed with either a pure self or pure pied-coloured form, the offspring are similar to, though not always exactly like, the coloured parent; provided, of course, that the albino is pure and is not carrying some colour or pattern determinant which is dominant to that of the coloured parent used.

This conflict arises not only from naturalization having been granted without the corresponding expatriation having been permitted, but also from the fact that birth on the soil was the leading determinant of nationality by feudal law, and still is so by the laws of England and the United States (jus soli), while the nationality of the father is its leading determinant in those countries which have accepted Roman principles of jurisprudence (jus sanguinis).

The physiography of the state is the evident determinant of its climate, fauna and flora.

a~ are the minors of the rth row of the determinant (7).

determinant a1 - I a2 O -I o 0 u an bn O O - - o o -I a n, from which point of view continuants have been treated by W.

Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange's investigation of the vibrations of a stretched string (see Lord Rayleigh, Theory of Sound, vol.

The idea, inasmuch as it is a law of universal mind, which in particular minds produces aggregates of sensations called things, is a "determinant" (iripas ixov), and as such is styled "quantity" and perhaps "number" but the ideal numbers are distinct from arithmetical numbers.

Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).

xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = o with the Hessian, or curve A = o, where A is the determinant formed with the second derived functions of u.

We might infer from this that the intellect, so judging, is itself the proper and complete determinant of the will, and that man, as a rational being, ought to aim at the realization of absolute good for its own sake.

With Price, again, he holds that rightness of intention and motive is not only an indispensable condition or element of the rightness of an action, but actually the sole determinant of its moral worth; but with more philosophical consistency he draws the inference - of which the English moralist does not seem to have dreamt - that there can be no separate rational principles for determining the " material " rightness of conduct, as distinct from its " formal " rightness; and therefore that all rules of duty, so far as universally binding, must admit of being exhibited as applications of the one general principle that duty ought to be done for duty's sake.

DETERMINANT, in mathematics, a function which presents itself in the solution of a system of simple equations.

Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

It is to be noticed that the resulting equation is a,b,c x= d,b,c,, ,, a' b' c' d'b' c' an, b", cn d", b", c" where the expression on the right-hand side is the like function with d, d', d" in place of a, a', a" respectively, and is of course also a determinant.

we have = a, 'l' a, b a', b'I a, b, c =ab', c' -Farb" a' b', c' b", c" b, c a", b" cn a, b, c, d a', b', c', d' a", b", c", d" a'" b in c'" d,n,, so on, the terms being all + for a determinant of an odd ord er, but alternately + and - for a determinant of an even order.

It is easy, by induction, to arrive at the general results: A determinant of the order n is the sum of the 1.2.3...n products which can be formed with n elements out of n 2 elements arranged in the form of a square, no two of the n elements being in the same line or in the same column, and each such product having the coefficient = unity.

The products in question may be obtained by permuting in every possible manner the columns (or the lines) of the determinant, and then taking for the factors the n elements in the dexter diagonal.

'This implies the theorem that a given arrangement can be derived from the primitive arrangement only by an odd number, or else only by an even number of interchanges, - a of which may be easily obtained from the theorem (in fact a particular case of the general one), an arrangement can be derived from itself only by an even number of interchanges.] And this being so, each product has the sign belonging to the corresponding arrangement of the columns; in particular, a determinant contains with the sign + the product of the elements in its dexter diagonal.

Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is =ab'c" - ab "c'+a'b "c - a'bc" - a"bc' - a"b'c. 3.

It further appears that a determinant is a linear function' of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or - according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs.

It at once follows that, if two columns are identical, or if two lines are identical, the value of the determinant is = o.

It may be added, that if the lines are converted into columns, and the columns into lines, in such a way as to leave the dexter diagonal unaltered, the value of the determinant is unaltered; the determinant is in this case said to be transposed.

By what precedes it appears that there exists a function of the n 2 elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient + 1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations.

Observe that the properties show at once that if any column is = o (that is, if the elements in the column are each = o), then the determinant is = o; and further, that if any two columns are identical, then the determinant is = o.

Reverting to the system of linear equations written down at the beginning of this article, consider the determinant ax+by+cz - d,b,c a' x+b' y+c'z - d', b', c" a"x+b"y+c"z - d", b", c" it appears that this is viz.

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