1. Let A and B are two nxn matrices, and I is a nxn identity matrix. If I + AB has an inverse, prove: (I + BA) = I–B (I + AB)'A.

a) Suppose that A and B are 4 × 4 matrices, det(A) = 2 and det((3ATB)−¹) = 81. Calculate det(B). b) Let A, B, and C be nxn matrices and suppose that ABC is invertible. Which of A, B, and C are necessarily invertible? Justify your answer.

For this exercise assume that the matrices are all nxn. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the equation Ax = b has at least one solution for each b in R, then the solution is unique for each b. Choose the correct answer below. A. The statement is true. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R", then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each b. OB. The statement is true, but only for x 0. By the Invertible Matrix Theorem, if Ax=b has at least one solution for each b in R, then the equation Ax = 0 does not only have the trivial solution. OC. The statement is false. By the…

Suppose A is a 2 × 2 matrix with real number entries such that AB=BA for all 2 × 2 matrices. What relationships must exist among the entries of A?