Chapter 4.6, Problem 36E

Let A be an $(m\times n)$ matrix, and let B be an $(n\times p)$ matrix, where the entries of A and B may be complex. Use Exercise 35 and the definition of AB to show that $\overline{AB}=\bar{A}\bar{B}.$ (By $\bar{A},$ we mean the matrix whose i j th entry is the conjugate of the i j th entry of A.) If A is a real matrix and x is an $(n\times 1)$ vector, show that $\overline{A\text{x}}=A\bar{\text{x}}.$