For an n x n matrix A, explain how to find each value: (a) The minor Mij of the entry aij: Take the determinant of the (n - 1) x (n - 1) matrix that is left after deleting the ith row and jth column. (b) The cofactor Cij of the entry aij: Cij = (-1)^(i+j) * Mij. (c) The determinant of A: IAI = a11C11 + a12C12 + ... + a1nC1n.
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Determine the cofactor matrix of the given matrix. A = ⌈ 3 2 -1 ⌉ ⌈ -2 0 4 ⌉ ⌈ 1 2 -5 ⌉ Find the minors of every element of the following matrices: For part (a): A = ⌈ 1 3 -2 ⌉ ⌈ 0 5 4 ⌉ ⌈ -3 4 -1 ⌉
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Theorem 2.2.2 An n % n matrix A is singular if and only if det(A) = 0 Proof The matrix A can be reduced to row echelon form with a finite number of row operations. Thus, U = E_{k}E_{k-1} ⋯ E_{1}A where U is in row echelon form and the E_{i}’s are all elementary matrices. It follows that det(U) = det(E_{k}E_{k-1} ⋯ E_{1}A) = det(E_{k}) det(E_{k-1}) ⋯ det(E_{1}) det(A) Since the determinants of the E_{i}’s are all nonzero, it follows that det(A) = 0 if and only if det(U) = 0. If A is singular, then U has a row consisting entirely of zeros, and hence det(U) = 0. If A is nonsingular, then U is triangular with 1’s along the diagonal and hence det(U) = 1. From the proof of Theorem 2.2.2, we can obtain a method for computing det(A). We reduce A to row echelon form. U = E_{k}E_{k-1} ⋯ E_{1}A If the last row of U consists entirely of zeros, A is singular and det(A) = 0. Otherwise, A is nonsingular and det(A) = [det(E_{k}) det(E_{k-1}) ⋯ det(E_{1})]^{-1} Actually, if A is nonsingular, it is simpler to reduce A to triangular form. This can be done using only row operations I and III. Thus, T = E_{m}E_{m-1} ⋯ E_{1}A and hence, det(A) = ± det(T) = ±t_{11}t_{22} ⋯ t_{nn} where the t_{ii}’s are the diagonal entries of T. The sign will be positive if row operation I has been used an even number of times and negative otherwise.
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