Chapter 6.CE, Problem 14CE

In Exercise $9-15$, give a brief answer.

1. a)

If $A$ and $B$ are nonzero $\left(n\times n\right)$ matrices such that $AB=\mathcal{O}$, then prove that both $A$ and $B$ are singular. [Hint: What would you conclude if either $A$ or $B$ were nonsingular?]

1. b)

Use part a) to prove that if $A$ is a singular $\left(n\times n\right)$ matrix, then $\text{Adj}\left(A\right)$ is also a singular matrix. [Hint: Consider the product $A\left[\text{Adj}\left(A\right)\right]$.]