How to find the inverse of the matrix

Question

How to find the inverse of the matrix? I am not asking for doing adjoint by determinant. Is there any other easy way to do it?

\begin{pmatrix} 1 & 0 & 0 &0 &0\\ 0 & 1 & 0 &0 &0\\ 0 & 0 & 1 &0 &0\\ 0 & 0 & 2 &1 &0\\ 0&0&3&0&1 \end{pmatrix}

Answer 1

You can "attach" an identity matrix $I_5$ use the Gauss-Jordan method until on the left-side you have the identity matrix and the right side you have an inverse matrix

Answer 2

Hint: Your matrix is $A=I+N$ where $N$ is nilpotent.

Solution:

We have $N^2=0$ and so $A(I-N)=I$. Therefore, $A^{-1}=I-N$.