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homework_4_solutions - MA 35100 HOMEWORK ASSIGNMENT#4...

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Unformatted text preview: MA 35100 HOMEWORK ASSIGNMENT #4 SOLUTIONS Problem 1. pg. 77; prob. 6 If possible, compute the matrix products in Exercises 1 through 13, using pencil and paper. a b c d d- b- c a Solution: We use Theorem 2.3.4 on page 73 of the text. The product is given by a b c d d- b- c a = a · d + b · (- c ) a · (- b ) + b · a c · d + d · (- c ) c · (- b ) + d · a = ad- bc ad- bc Remark: Given a 2 × 2 matrix A , we denote the cofactor of A as the 2 × 2 matrix cof( A ) = d- b- c a where A = a b c d . The previous exercise shows that we have the product A · cof( A ) = det( A ) I 2 in terms of the 2 × 2 identity matrix I 2 . Problem 2. pg. 77; prob. 7 If possible, compute the matrix products in Exercises 1 through 13, using pencil and paper. 1- 1 1 1 1- 1- 2 1 2 3 3 2 1 2 1 3 Solution: Again, we use Theorem 2.3.4 on page 73 of the text: 1- 1 1 1 1- 1- 2 1 2 3 3 2 1 2 1 3 = 1 · 1 + 0 · 3 + (- 1) · 2 1 · 2 + 0 · 2 + (- 1) · 1 1 · 3 + 0 · 1 + (- 1) · 3 · 1 + 1 · 3 + 1 · 2 · 2 + 1 · 2 + 1 · 1 · 3 + 1 · 1 + 1 · 3 1 · 1 + (- 1) · 3 + (- 2) · 2 1 · 2 + (- 1) · 2 + (- 2) · 1 1 · 3 + (- 1) · 1 + (- 2) · 3 = - 1 1 5 3 4- 6- 2- 4 Problem 3. pg. 77; prob. 8 1 If possible, compute the matrix products in Exercises 1 through 13, using pencil and paper. 0 1 0 0 0 1 0 0 Solution: Once more we use Theorem 2.3.4 on page 73 of the text: 0 1 0 0 0 1 0 0 = · 0 + 1 · 0 0 · 1 + 1 · · 0 + 0 · 0 0 · 1 + 0 · = 0 0 0 0 Problem 4. pg. 78; prob. 55 In Exercises 55 through 64, find all matrices X that satisfy the given matrix equation. 1 2 2 4 X = 0 0 0 0 . Solution: The product of an n × p matrix by a p × m matrix is an n × m matrix. As the matrix on the left is a n × p = 2 × 2 matrix and the matrix on the right is a n × m = 2 × 2 matrix, we must have n = m = p = 2. We denote the 2 × 2 matrix X as X = a 11 a 12 a 21 a 22 so that 1 2 2 4 X = a 11 + 2 a 21 a 12 + 2 a 22 2 a 11 + 4 a 21 2 a 12 + 4 a 22 ....
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homework_4_solutions - MA 35100 HOMEWORK ASSIGNMENT#4...

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