Find the inverseof the following matrix:

$$A= \begin{bmatrix}
0 & 1 & 0 & 0 &0 \\
1 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0&1 \\
0 & 0 & 1 & 0 &0 \\0&0&0&1&0 
\end{bmatrix}$$

My attempt : I think the inverse will not exist because $\det A=0 $

Is it True ?

Any hints/solution will be appreciated.