Question

For where it says "choose" is either is/is not or are/are not

Answer 1

Answer:

IS NOT; ARE NOT

Step-by-step explanation:

Given: $\[ \begin{bmatrix} \frac{1}{4} & \frac{1}{4}\\ \\-1 & \frac{-1}{2} \end{bmatrix}\]$

and $\[A = \begin{bmatrix} \frac{1}{4} & \frac{1}{4} \\\\ -1 & \frac{-1}{2} \end{bmatrix}\]$

We say two matrices $A$ and $B$ are inverses of each other when $AB = BA = I$ where $I$ is the identity matrix.

$\[I = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\]$

So, for $X$ and $A$ to be inverses of each other, we should have $AX = XA = I$.

Let us calculate $XA$.

$\[\begin{bmatrix} -2 & -1 \\ 8 & 2 \end{bmatrix}\]$$\[\begin{bmatrix} \frac{1}{4} & \frac{1}{4} \\\\ -1 & \frac{-1}{2}\end{bmatrix}\]$$=$ $\[\begin{bmatrix}\frac{1}{2} & 0 \\0 & 0 \end{bmatrix}\]$

This is clearly not equal to the identity matrix. So we conclude that the matrices are not inverses of each other.