$($a$)$ Prove that if A and $B$ are invertible $nn$ matrices, then $AB$ is invertible and $(AB{)}^{-1}=B^{-1}{A}^{-1}.$

$5$

$($b$)$ If possible, find the inverse of each of the following two matrices:

$A=([3,-1],[-2,2]),B=([3,-1],[-6,2])$

$5$

$($c$)$ Show that the matrix

$A=([1,2,0],[3,-1,2],[-2,3,-2])$

has no inverse.

$5$

$($d$)$ Give an example of two matrices A and $B$ such that $AB=0$ with $A0$ and $B0.$

$10$