# Determinant Sentence Examples

Consideration of the definition of the

**determinant**shows that the value is unaltered when the suffixes in each element are transposed.Indeed, while many diseases and health conditions are imprinted in our genetic code, our environment is a critical

**determinant**of its unfolding.If the

**determinant**is transformed so as to read by columns as it formerly did by rows its value is unchanged.No member of a

**determinant**can involve more than one element from the first row.The adjoint

**determinant**is the (n - I) th power of the original**determinant**.AdvertisementResultant 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.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.Y ...a n v, the summation being for all permutations of the n numbers, is called the

**determinant**of the n 2 quantities.Each row as well as each column supplies one and only one element to each member of the

**determinant**.AdvertisementHence 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**.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**.This

**determinant**and that associated with Aik are termed corresponding determinants.AdvertisementWhen 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.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**.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),...AdvertisementIf 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.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.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.AdvertisementAnd 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.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).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.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**.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.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.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.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.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**.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**.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.The fermion

**determinant**is calculated by summing over the resulting complex eigenvalues.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.A matrix has in many parts of mathematics a signification apart from its evaluation as a

**determinant**.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.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.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.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.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).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.The

**determinant**is usually written all a12 a13.If any two rows or any two columns of a

**determinant**be identical the value of the**determinant**is zero.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.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 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.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.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.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.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.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.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 +.The subject matter of the text is, obviously, a crucial

**determinant**of the role that spatial inferences play in understanding it.Coupled with the 24-hour pH probe study, the test becomes the best

**determinant**of GERD because it actually monitors how often the patient has reflux into the esophagus during a full day.Youth gangs are bound by a common ethnicity, race, social class, or other

**determinant**and employ distinctive symbols, including style and color of dress, hand signs, tattoos, and graffiti.Motivation to change risk behaviors is another

**determinant**of prevention and affects whether a person acts on his or her knowledge of the transmission and prevention of HIV.Behavioral skills for engaging in specific prevention behaviors are a third

**determinant**of prevention; it affects whether a knowledgeable, highly motivated person will be able to change his or her behavior to prevent HIV.Bleeding from the nose is the obvious

**determinant**of a nosebleed.A good

**determinant**in making your decision is to remind yourself that if it sounds too good to be true, it probably is.This helps preserve muscle mass and keep your natural metabolism going, which is a key

**determinant**to your progress.We thus obtain for the product a

**determinant**of order n.While cost is always an important

**determinant**when shopping for insurance, it is also necessary to keep in mind just what type of financial risk you're attempting to shield yourself from.This rate is used as a

**determinant**for a company's possible growth.In the original Star Trek, the hierarchy of officers was a key

**determinant**of the big three.