Answer
The given matrices are inverses of each other.
Work Step by Step
We know that when two matrices $P$ and $Q$ are two $n\times n$ matrices so that $PQ=QP=I_n$, then $P$ and $Q$ are inverses of each other.
Here, we have: $P=\begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix}$ and $Q=\begin{bmatrix} 3 & -7 \\ -2 & 5 \end{bmatrix} $ .
As we can see that both matrices of the same dimensions, that is, $2 \times 2$ or, $n=2$. Thus, we can multiply them easily.
$PQ=\begin{bmatrix} 5 & 7 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 3 & -7 \\ -2 & 5 \end{bmatrix}\\ =\begin{bmatrix} (5)(3) +(7)(-2) & (5)(-7) +(7)(5) \\ (2)(3) +(3)(-2)& (2)(-7) +(3)(5) \end{bmatrix} \\\\=\begin{bmatrix} 1 & 0 \\ 0& 1 \end{bmatrix} $
This implies that the given matrices are inverses of each other because $PQ= I_2$, that is, their products shows an identity matrix.
