# Determinant Sentence Examples

- Consideration of the definition of the
**determinant**shows that the value is unaltered when the suffixes in each element are transposed. - If any two rows or any two columns of a
**determinant**be identical the value of the**determinant**is zero. - 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. - If the
**determinant**is transformed so as to read by columns as it formerly did by rows its value is unchanged. - 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. - Bezout's method gives the resultant in the form of a
**determinant**of order m or n, according as m is n. - 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. - 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 leading member of the
**determinant**is alla22a33ï¿½ï¿½ï¿½ann, and corresponds to the principal diagonal of the matrix. - No member of a
**determinant**can involve more than one element from the first row. - 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 correspondingit 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**determinants****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. - 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). - 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. - 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. - When a skew symmetric
**determinant**is of even degree it is a perfect square. - 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?- = (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). - 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),... - The
**determinant**is usually written all a12 a13. - 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 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. - 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. - 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 bothso as to read by columns as they formerly did by rows.**determinants** - 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. - Hence the product
**determinant**has the principal diagonal elements each equal to A and the remaining elements zero. - 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. - A skew symmetric
**determinant**has a,. - 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. - +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. - 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. - 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. - 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.