For an n x n matrix A, explain how to find each value. (a) The minor M of the entry a. O Take the determinant of the (n- 1) x (n - 1) matrix that is left after deleting the ith row and ith column. O Take the determinant of the (n - 1) x (n - 1) matrix that is left after deleting the ith row and jth column. Take the determinant of the (n - 1) x (n - 1) matrix that is left after deleting the jth row and jth column. O Take the determinant of the n x n matrix that is left after deleting the ith row and jth column. Take the determinant of the n x n matrix. (b) The cofactor C; of the entry ajj. O If i +j is negative, then Cij = -Mij. If i + j is positive, then C; = Mij. O Cij = Mij %3D O If i +j is odd, then Cj = Mij. If i +j is even, then C; = -Mij- %3D If i +j is odd, then Cij = -Mij. If i +j is even, then Ci; = Mij- %3D Cij = -Mij (c) The determinant of A. O JAI = a11C12 + A12C11 + ... + a1„C1n O IAI = a11C11 + a12C12 + + a1,Cin O JAI = a11C11 - 212C12 + . .. - ainCin O IAI = a11C11 - A12C12 - ... - ainCin O IAI = a11C11 + a22C22 + ... + annCnn

For an n x n matrix A, explain how to find each value. (a) The minor M of the entry a. O Take the determinant of the (n- 1) x (n - 1) matrix that is left after deleting the ith row and ith column. O Take the determinant of the (n - 1) x (n - 1) matrix that is left after deleting the ith row and jth column. Take the determinant of the (n - 1) x (n - 1) matrix that is left after deleting the jth row and jth column. O Take the determinant of the n x n matrix that is left after deleting the ith row and jth column. Take the determinant of the n x n matrix. (b) The cofactor C; of the entry ajj. O If i +j is negative, then Cij = -Mij. If i + j is positive, then C; = Mij. O Cij = Mij %3D O If i +j is odd, then Cj = Mij. If i +j is even, then C; = -Mij- %3D If i +j is odd, then Cij = -Mij. If i +j is even, then Ci; = Mij- %3D Cij = -Mij (c) The determinant of A. O JAI = a11C12 + A12C11 + ... + a1„C1n O IAI = a11C11 + a12C12 + + a1,Cin O JAI = a11C11 - 212C12 + . .. - ainCin O IAI = a11C11 - A12C12 - ... - ainCin O IAI = a11C11 + a22C22 + ... + annCnn

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Determinants

Section3.1: The Determinants Of A Matrix

Problem 70E: The determinant of a 22 matrix involves two products. The determinant of a 33 matrix involves six...

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