$(20$ points$)$

Determine whether the statement is true or false. Briefly explain your answers.

$($a$)$ If $AB=AC,$ then $B=C.$

$($b$)$ If $A=([3,5],[1,2]),$ then $A^{-1}=([2,5],[1,-3]).$

$($c$)$ If $A=([1,3],[-5,2]),$ then $A^2=([1,9],[25,4]).$

$($d$)$ If $A$ is a $64$ matrix and $B$ is a $45$ matrix, then $AB$ has $16$ entries.

$($e$)$ If A and $B$ are $22$ matrices such that $AB=([0,0],[0,0]),$ then $A=([0,0],[0,0])$ or $B=([0,0],[0,0]).$

Solution: