6-2 Matrix Multiplication, Inverses and Determinants
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1 Find AB and BA, if possible. 4. A = B = A = ; B = A is a 2 1 matrix and B is a 1 4 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 1
2 8. A = B = A = ; B = A is a 2 3 matrix and B is a 4 2 matrix. Because the number of columns of A is not equal to the number of rows of B, AB is undefined. B is a 4 2 matrix and A is a 2 3 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. Write each system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. 12. Write the system matrix in form AX = B. Make sure you align the variables. For the first matrix, the first column should include x 1, the second column x 2, and the third column x 3. The column matrix and the matrix of constant terms should be listed in order. esolutions Manual - Powered by Cognero Page 2
3 terms should be listed in order. = Write the augmented matrix. Attach the matrix of constant terms to the end of the 3 3 matrix. Use Gauss-Jordan elimination to solve the system. First, use elementary row operations to transform A into I. The solution of the equation is given by X. esolutions Manual - Powered by Cognero Page 3
4 The solution of the equation is given by X. Therefore, the solution is (3, 0.5, 8). 16. Write the system matrix in form AX = B. Make sure you align the variables. For the first matrix, the first column should include x 1, the second column x 2, and the third column x 3. The column matrix and the matrix of constant terms should be listed in order. = Write the augmented matrix. Attach the matrix of constant terms to the end of the 3 3 matrix. Use Gauss-Jordan elimination to solve the system. First, use elementary row operations to transform A into I. esolutions Manual - Powered by Cognero Page 4
5 The solution of the equation is given by X. Therefore, the solution is ( 1, 24, 9). esolutions Manual - Powered by Cognero Page 5
6 Determine whether A and B are inverse matrices. 20. A = B = A = ; B = If A and B are inverse matrices, then AB = BA = I. Because AB = BA = I, B = A 1 and A = B A = B = A = ;B = If A and B are inverse matrices, then AB = BA = I. AB I, so A and B are not inverses. esolutions Manual - Powered by Cognero Page 6
7 Find A -1, if it exists. If A -1 does not exist, write singular. 28. 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 2 columns cannot become the identity matrix. Therefore, A is singular. 32. A = Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. 36. A row of 0s has been formed, so the first 3 columns cannot become the identity matrix. Therefore, A is singular. Find the determinant of each matrix. Then find the inverse of the matrix, if it exists. esolutions Manual - Powered by Cognero Page 7
8 Find the determinant. det(a) = A = = ad bc Because the determinant is not 0, the matrix is invertible. Find the inverse matrix. Confirm that AA 1 = A 1 A = I. esolutions Manual - Powered by Cognero Page 8
9 esolutions Manual - Powered by Cognero Page 9
10 40. Find the determinant. det(a) = A = = Because the determinant is not 0, the matrix is invertible. Use a graphing calculator to find the inverse matrix. Use the Frac feature under the MATH menu to write the inverse using fractions. Therefore, A 1 =. esolutions Manual - Powered by Cognero Page 10
11 44. Find the determinant. det(a) = A = = Because the determinant is not 0, the matrix is invertible. Use a graphing calculator to find the inverse matrix. Use the Frac feature under the MATH menu to write the inverse using fractions. Therefore, A 1 =. esolutions Manual - Powered by Cognero Page 11
12 Find the area A of each triangle with vertices (x 1, y 1 ), (x 2, y 2 ), and (x 3, y 3 ), by using A =, where X is. 48. Find the determinant of each matrix. 52. esolutions Manual - Powered by Cognero Page 12
13 56. HORSES The owner of each horse stable buys bales of hay and bags of feed each month. In May, hay cost $2.50 per bale and feed cost $7.95 per bag. In June, the cost per bale of hay was $3.00 and the cost per bag of feed was $6.75. a. Write a matrix X to represent the bales of hay i and bags of feed j that are bought monthly by each stable. b. Write a matrix Y to represent the costs per bale of hay and bags of feed for May and June. c. Find the product YX and label its rows and columns. d. How much more were the total costs in June for Fairwind Farms than the total costs in May for Galloping Hills? a. Each column will represent a stable. X = b. Let May be the 1 st row and June be the 2 nd row. Y = c. d. Fairwind Farms (June): 135(3.00) + 16(6.75) = $513 Galloping Hills (May): 45(2.50) + 5(7.95) = $ = $ esolutions Manual - Powered by Cognero Page 13
14 Evaluate each expression. 60. AB + CB esolutions Manual - Powered by Cognero Page 14
15 Solve each equation for X, if possible. 64. DA = 7X esolutions Manual - Powered by Cognero Page 15
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