WRONG APPROACH
we know that $A^{-1} = \frac{1}{|A|} (adj(A))$
Multiply by $A$ on both sides
$AA^{-1} = \frac{1}{|A|}A(adj(A))$ $\Leftrightarrow$ $|A|I = A(adj(A))$
from the matrix given, we conclude that $|A|= \frac{1}{5^{3}}$
We want to find the value of $\vert(5A)^{-1}\vert$.
$\vert(5A)^{-1}\vert =\frac{1}{\vert 5A\vert}= $ $ \frac{1}{5^{3}\vert A \vert}$ chip in the value of $\vert A\vert$ from above.
Therefore $ \vert (5A)^{-1} \vert = 1$
CORRECT APPROACH
There is a slight in the question , in RHS that is not a determinant but a matrix. So from now on in this solution we will consider it as a matrix.
$A^{-1} = \frac{1}{|A|} (adj(A))$
Multiply by $A^{-1}$ on both sides
$AA^{-1} = \frac{1}{|A|}A(adj(A))$ $\Leftrightarrow$ $|A|I = A(adj(A))$
From the given matrix, we conclude that$|A|I=\frac{1}{5}I \Rightarrow$ $|A|=\frac{1}{5}$
We want to find the value of $\vert(5A)^{-1}\vert$.
$\vert(5A)^{-1}\vert =\frac{1}{\vert 5A\vert}= $ $ \frac{1}{5^{3}\vert A \vert}$ chip in the value of $\vert A\vert$ from above.
Therefore $ \vert (5A)^{-1} \vert = \frac{1}{5^2}$
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