Explanation Given: The matrices A = [ 1 3 − 3 2 4 1 − 1 3 2 ] , B = [ 1 2 1 2 − 1 3 ] , C = [ 2 3 − 1 2 3 − 2 ] , and D = [ − 2 3 1 − 3 2 2 ] . Procedure used: Finding Inverse of a Square Matrix: 1. Form an augmented matrix [ A | I ] where a matrix A is placed along with a same dimension identity matrix I separated by a line. 2. Apply row operations on [ A | I ] in order to obtain another augmented matrix [ I | B ] where I is again an identity matrix and B is an inverse of A. 3. If the reduction process yields a row of zeros in the left half of the augmented matrix, then the inverse of A does not exist. Calculation: First obtain the determinant of the matrix A to check whether the matrix A is invertible or not. The determinant of the matrix A is as follows. | A | = 1 ( 4 ⋅ 2 − 3 ⋅ 1 ) − 3 [ 2 ⋅ 2 − ( − 1 ) ⋅ 1 ] + ( − 3 ) [ 2 ⋅ 3 − ( − 1 ) ⋅ 4 ] = ( 8 − 3 ) − 3 ( 4 + 1 ) − 3 ( 6 + 4 ) = 5 − 15 − 30 = − 40 Since, the determinant | A | ≠ 0 ; hence the matrix A is nonsingular and invertible. Now, use the method of finding inverse of the matrix and obtain the A − 1 as follows. The augment matrix [ A | I ] is [ A | I ] = [ 1 3 − 3 1 0 0 2 4 1 0 1 0 − 1 3 2 0 0 1 ] . Perform the row operations R 2 − 2 R 1 → R 2 and R 3 + R 1 → R 3 , and obtain the row reduced equivalent matrix as, [ 1 3 − 3 1 0 0 0 − 2 7 − 2 1 0 0 6 − 1 1 0 1 ] Perform the row operations R 3 + 3 R 2 → R 3 and obtain the equivalent row reduced matrix as, [ 1 3 − 3 1 0 0 0 − 2 7 − 2 1 0 0 0 20 − 5 3 1 ] Again perform row operation 1 20 R 3 → R 3 and obtain the equivalent row reduced matrix as, [ 1 3 − 3 1 0 0 0 − 2 7 − 2 1 0 0 0 1 − 5 20 </

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Chapter 7, Problem 21RE

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To perform: The matrix operation A−1, if possible.

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Chapter 7, Problem 20RE

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Chapter 7 Solutions

EBK COLLEGE ALGEBRA IN CONTEXT

Ch. 7.1 - In Exercises 14, solve the systems of equations....Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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To perform: The matrix operation A − 1 , if possible.

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