# 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. - 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. - 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. - - 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. - This
**determinant**and that associated with Aik are termed corresponding.**determinants** - Since A lk is a
**determinant**we similarly obtain Alk = a21Alk+ï¿½ ï¿½ ï¿½ +a2,k-iAl,k +a2,k+lAl,k+ ï¿½ï¿½ï¿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 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. - 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. - 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. - 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. - 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. - 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. - 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. - 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),... - 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 .> - 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. - 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. - Hence the product
**determinant**has the principal diagonal elements each equal to A and the remaining elements zero. - It is easy to see that the adjoint
**determinant**is also 'symmetrical, viz. - A skew symmetric
**determinant**has a,. - +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. - 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. - 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.