Chapter 2 Determinants by Cofactor Expansion Evaluating Determinants

Chapter 2 • Determinants by Cofactor Expansion • Evaluating Determinants by Row Reduction • Properties of the Determinants, Cramer’s Rule 9/15/2020 1

Minor and Cofactor Let A be a 2 x 2 matrix Then ad-bc is called the determinant of the matrix A, and is denoted by The symbol det(A). We will extend the concept of a determinant to square matrices of all orders. 9/15/2020 2

Minor and Cofactor Definition If A is a square matrix, then • The minor of entry aij, denoted Mij, is defined to be the determinant of the submatrix that remains after the ith row and jth column are deleted from A. • The number (-1)i+j Mij is denoted by Cij and is called the cofactor of entry aij Remark Note that Cij = Mij and the signs checkerboard pattern: 9/15/2020 in the definition of cofactor form a 3

Example: Let . Find the minor and cofactor of a 12, and a 23 Solution: The minor of entry a 12 is The cofactor of a 12 is Similarly, the minor of entry a 23 is The cofactor of a 23 is 9/15/2020 4

Cofactor Expansion Theorem 2. 1. 1 (Expansions by Cofactors) The determinant of an n n matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i, j n det(A) = a 1 j. C 1 j + a 2 j. C 2 j +… + anj. Cnj (cofactor expansion along the jth column) and det(A) = ai 1 Ci 1 + ai 2 Ci 2 +… + ain. Cin (cofactor expansion along the ith row) 9/15/2020 5

Example: Let . Evaluate det (A) by cofactor expansion along the first column of A. Solution: 9/15/2020 6

Determinant of an Upper Triangular Matrix Theorem If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix; that is, det(A)=a 11 a 22…ann. Arrow Technique for determinants of 2 x 2 and 3 x 3 matrices. 9/15/2020 7

Section 2. 3 Properties of Determinants; Cramer’s Rule Definition (Adjoint of a Matrix) If A is any n n matrix and Cij is the cofactor of aij, then the matrix is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj(A) 9/15/2020 8

Example: Let Solution: The cofactors of A are: . Find the adjoint of A. C 11 = 12, C 12 = 6, C 13 = -16, C 21 = 4, C 22 = 2, C 23 = 16, C 31 = 12, C 32 = -10, C 33 = 16 The matrix of cofactors is and adjoint of A is 9/15/2020 9

Theorems Theorem 2. 1. 2 (Inverse of a Matrix using its Adjoint) If A is an invertible matrix, then Example: Use theorem 2. 1. 2. to find the inverse of Solution: det (A)= a 11 C 11+a 12 C 12+a 13 C 13 = 3(12)+2(6)-1(-16)= 64 Thus, 9/15/2020 10

Cramer’s Rule Theorem 2. 1. 4 (Cramer’s Rule) If Ax = b is a system of n linear equations in n unknowns such that det(A) 0 , then the system has a unique solution. This solution is where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix 9/15/2020 11

Example: Use Cramer’s rule to solve Solution: Therefore, 9/15/2020 12

Properties of the Determinant Function Suppose that A and B are nxn matrices and k is any scalar. We obtain But There is no simple relationship exists between det(A), det(B), and det(A+B) in general. In particular, det(A+B) is usually not equal to det(A) + det(B). Example: Consider We have det (A)=1, det (B)=8, and det(A+B)=23. Thus 9/15/2020 13

Properties of the Determinants Sum Theorem 2. 3. 1 Let A, B, and C be n n matrices that differ only in a single row, say the r-th, and assume that the r-th row of C can be obtained by adding corresponding entries in the r-th rows of A and B. Then det(C) = det(A) + det(B) The same result holds for columns. Example 9/15/2020 14

Properties of the Determinants Multiplication Lemma 2. 3. 2 If B is an n n matrix and E is an n n elementary matrix, then det(EB) = det(E) det(B) Theorem 2. 3. 3 (Determinant Test for Invertibility) A square matrix A is invertible if and only if det(A) 0 9/15/2020 15

Theorem 2. 3. 4 If A and B are square matrices of the same size, then det(AB) = det(A) det(B) Theorem 2. 3. 5 If A is invertible, then 9/15/2020 16

Theorem 2. 3. 6 (Equivalent Statements) If A is an n n matrix, then the following are equivalent • A is invertible. • Ax = 0 has only the trivial solution • The reduced row-echelon form of A as In • A is expressible as a product of elementary matrices • Ax = b is consistent for every n 1 matrix b • Ax = b has exactly one solution for every n 1 matrix b • det(A) 0 9/15/2020 17