# solhw7 - Solutions to Homework#7 MAT1341A Question 1(2 2 =...

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Unformatted text preview: Solutions to Homework #7, MAT1341A Question # 1: (2+2 = 4 points) Simplify the following expressions, if possible. Assume all matrices are n × n and invertible. (a) ( ABA- 1 A ) T B- 1 Solution: ( ABA- 1 A ) T B- 1 = ( ABI ) T B- 1 = ( AB ) T B- 1 = B T A T B- 1 (b) (( AB ) T )- 1 Solution: (( AB ) T )- 1 = ( B T A T )- 1 = ( A T )- 1 ( B T )- 1 Question # 2: (3+3+3+3+3 = 15 points) Find the inverses of each of the following matrices, if they exist. If they are not invertible, explain why not. If they are invertible, check your answer by multiplying AA- 1 . (Continued on next page.) (a) A = 6 1 20 1 3 4- 3 8 Solution: 6 1 20 1 1 3 1 4- 3 8 1 ----→ R 1 R 2 1 3 1 6 1 20 1 4- 3 8 1 R 2 → R 2- 6 R 1-------→ R 3 → R 3- 4 R 1 1 3 1 1 2 1- 6- 3- 4- 4 1 -------→ R 3 → R 3 +3 R 2 1 3 1 1 2 1- 6 2 3- 22 1 -----→ R 3 → 1 2 R 3 1 3 1 1 2 1- 6 1 3 2- 11 1 2 R 2 → R 2- 2 R 3-------→ R 1 → R 1- 3 R 3 1- 9 2 34- 3 2 1- 2 16- 1 1 3 2- 11 1 2 On the left hand side, we got I , so the inverse exists and it’s on the right hand side. A- 1 = - 9 2 34- 3 2- 2 16- 1 3 2- 11 1 2 Check: 6 1 20 1 3 4- 3 8 - 9 2 34- 3 2- 2 16- 1 3 2- 11 1 2 = 1 1 1 - 9 2 34- 3...
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solhw7 - Solutions to Homework#7 MAT1341A Question 1(2 2 =...

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