Answer:
IS NOT; ARE NOT
Step-by-step explanation:
Given: [tex]\[ \begin{bmatrix} \frac{1}{4} & \frac{1}{4}\\ \\-1 & \frac{-1}{2} \end{bmatrix}\][/tex]
and [tex]\[A = \begin{bmatrix} \frac{1}{4} & \frac{1}{4} \\\\ -1 & \frac{-1}{2} \end{bmatrix}\][/tex]
We say two matrices [tex]$ A $[/tex] and [tex]$ B $[/tex] are inverses of each other when [tex]$ AB = BA = I $[/tex] where [tex]$ I $[/tex] is the identity matrix.
[tex]\[I = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\][/tex]
So, for [tex]$ X $[/tex] and [tex]$ A $[/tex] to be inverses of each other, we should have [tex]$ AX = XA = I $[/tex].
Let us calculate [tex]$ XA $[/tex].
[tex]\[\begin{bmatrix} -2 & -1 \\ 8 & 2 \end{bmatrix}\][/tex][tex]\[\begin{bmatrix} \frac{1}{4} & \frac{1}{4} \\\\ -1 & \frac{-1}{2}\end{bmatrix}\][/tex][tex]$ = $[/tex] [tex]\[\begin{bmatrix}\frac{1}{2} & 0 \\0 & 0 \end{bmatrix}\][/tex]
This is clearly not equal to the identity matrix. So we conclude that the matrices are not inverses of each other.
