6-2 Matrix Multiplication, Inverses and Determinants
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1 Find AB and BA, if possible. 1. A = A = ; A is a 1 2 matrix and B is a 2 2 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find the sum of the products of the entries in row 1 of A and column 1 of B. Follow the same procedure for row 2 column 1 of AB. Because the number of columns of B is not equal to the number of rows of A, BA is undefined. 3. A = A = ; A is a 1 2 matrix and B is a 2 3 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find the sum of the products of the entries in row 1 of A and column 1 of B. Follow the same procedure for row 2 column 1 of AB and the remaining entry. Because the number of columns of B is not equal to the number of rows of A, BA is undefined. esolutions Manual - Powered by Cognero Page 1
2 5. A = A = ; A is a 3 1 matrix and B is a 2 3 matrix. Because the number of columns of A is not equal to the number of rows of B, AB is undefined. B is a 2 3 matrix and A is a 3 1 matrix. Because the number of columns of B is equal to the number of rows of A, BA exists. To find the first entry of BA, find the sum of the products of the entries in row 1 of B and column 1 of A. Follow the same procedure for row 2 column 1 of BA and the remaining entries. esolutions Manual - Powered by Cognero Page 2
3 7. A = A = ; A is a 2 2 matrix and B is a 2 3 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find the sum of the products of the entries in row 1 of A and column 1 of B. Follow the same procedure for row 2 column 1 of AB and the remaining entries. Because the number of columns of B is not equal to the number of rows of A, BA is undefined. esolutions Manual - Powered by Cognero Page 3
4 Determine whether A and B are inverse matrices. 19. A = A = ; If A and B are inverse matrices, then A BA = I. Because A BA = I, A 1 and A = B A = A = ; If A and B are inverse matrices, then A BA = I. AB I, so A and B are not inverses. esolutions Manual - Powered by Cognero Page 4
5 23. A = A = ; If A and B are inverse matrices, then A BA = I. Because A BA = I, A 1 and A = B 1. esolutions Manual - Powered by Cognero Page 5
6 25. A = A = ; If A and B are inverse matrices, then A BA = I. 27. A = Because A BA = I, A 1 and A = B 1. Find A -1, if it exists. If A -1 does not exist, write singular. Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. A row of 0s has been formed, so the first 2 columns cannot become the identity matrix. Therefore, A is singular. esolutions Manual - Powered by Cognero Page 6
7 29. A = Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. The first two columns are the identity matrix. Therefore, A is invertible and A 1 =. Confirm that AA 1 = A 1 A = I. 31. A = esolutions Manual - Powered by Cognero Page 7
8 Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. The first three columns are the identity matrix. Therefore, A is invertible and A 1 =. Confirm that AA 1 = A 1 A = I. esolutions Manual - Powered by Cognero Page 8
9 esolutions Manual - Powered by Cognero Page 9
10 33. A = Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. A row of 0s has been formed, so the first 3 columns cannot become the identity matrix. Therefore, A is singular. esolutions Manual - Powered by Cognero Page 10
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