$($a$)$ Let A and $B$ be two real matrices such that the product $AB$ is defined. Show that if A has two identical rows, then the corresponding two rows of $AB$ are also identical.

$5$

$($b$)$ Let $i=\sqrt[\phantom{2}]{-1}$inC be the imaginary unit and let

$A=([i,0],[0,i]),B=([0,-i],[i,0])$

Find $A^{100}$ and $B^{100}.$

$8$

$($c$)$ Show that no real $22$ matrices $A,Bin{M}_{22}(R)$ exist that satisfy the matrix equation

$AB-BA=([1,0],[0,1])$

$[7]$