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where the last form puts into evidence the elements of A, upon which the number depends. determinants of order n 1 and n 2, we have explicitly    a11 a12    ja11 j a11  a21 a22  a11 a22 a12 a21
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For larger n, the analogous expressions become very cumbersome, and they are usually avoided by use of Laplace s expansion theorem (see below). What is important is that the determinant is de ned in such a way that det AB det A det B for any two n n matrices A and B. Two other basic properties are:
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det A det A Finally, det A 6 0 if and only if A is nonsingular.
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EXAMPLE B3 Verify the deteminant multiplication rule for ! ! 1 4 2 9 A B 3 2 1  We have ! ! ! 1 4 2 9 2 9 4 AB 3 2 1  4 27 2    2 9 4     4 27 2  2 27 2 9 4 4 90 20   1 4    3 2  1 2 4 3 10    2 9     1   2  9 1 9 2
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Laplace s Expansion Theorem The minor, Mij , of the element aij of a determinant of order n is the determinant of order n 1 obtained by deleting the row and column containing aij . The cofactor, ij , of the element aij is de ned as ij 1 i j Mij Laplace s theorem states: In the determinant of a square matrix A, multiply each element in the pth row (column) by the cofactor of the corresponding element in the qth row (column), and sum the products. Then the result is 0, for p 6 q; and det A, for p q. It follows at once from Laplace s theorem that if A has two rows or two columns the same, then det A 0 (and A must be a singular matrix).
Matrix Inversion by Determinants; Cramer s rule Laplace s expansion theorem can be exhibited as a matrix multiplication, as follows: 2 a11 a12 a22 ... an2 a13 a23 ... ... a1n 32 11 21 22 31 32 ... ... n1 3 6a 6 21 6 4 ... an1 2 a2n 76 12 76 76 . . . 54 . . . n2 7 7 7 ... 5
11 6 6 12 6 4 ... 2
... ... ... ... ... 1n 2n 3n . . . nn an3 . . . ann 32 3 21 31 . . . n1 a11 a12 a13 . . . a1n 22 32 . . . n2 76 a21 a22 a23 . . . a2n 7 76 7 76 7 . . . . . . . . . . . . 54 . . . . . . . . . . . . . . . 5 an1 3 7 7 7 5 an2 an3 . . . ann
1n 2n 3n . . . nn det A 0 0 ... 0 det A ... 0 0 ... ... ... 0 ... 0 ... det A
6 0 6 6 4 ... 0 or
A adj A adj A A det A I
where adj A  ji is the transposed matrix of the cofactors of the aij in the determinant of A, and I is the n n unit matrix. If A is nonsingular, one may divide through by det A 6 0, and infer that A 1 1 adj A det A
This means that the unique solution of the linear system Y AX is   1 adj A Y X det A which is Cramer s rule in matrix form. The ordinary, determinant form is obtained by considering the rth row r 1; 2; . . . ; n of the matrix solution. Since the rth row of adj A is 1r 2r 3r . . . nr we have:
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