Chapter 2.2, Problem 9E

In Exercises 9 and 10, mark each statement True or False. Justify each answer.

9. a. In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.

b. If A and B are n x n and invertible, then A⁻¹ B⁻¹ is the inverse of AB.

c. If A = $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and ab − cd ≠ 0, then A is invertible.

d. If A is an invertible n × n matrix, then the equation Ax = b is consistent for each b in ℝⁿ.

e. Each elementary matrix is invertible.