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Homework for §3.5 Solve the system of equations A~x = ~b for the given choices of matrix A and vector ~b by first finding A−1 and then computing ~x = A−1~b. 3 −1 1 ~ . , b= 1. A = 3 2 1 1 0 4 3 ~ 2 −1 3 2 . 2. A = , b= 0 2 −2 0 2 1 0 0 2 6 0 0 1 1 ~ 3. A = 0 1 0 0 , b = 4 . 1 0 0 0 3 ......................................................................................... 4. Solve the matrix equation AX = B, where 2 1 −2 2 3 and B = A= 3 4 6 −1 9 by computing X = A−1 B. 5. Suppose that A is an n × n matrix such that A~x = ~x for every n × 1 vector ~x. Explain why it must be that A = I. (Hint: Saying “because the identity is the only matrix that has this property” is not an explanation.)