The matrix row-reduces to
\begin{equation*}
\begin{bmatrix}
\leading{1}& 0 & 0 & 0\\
0 & \leading{1} & 0 & 0\\
0 & 0 & \leading{1} & 0\\
0 & 0 & 0 & \leading{1}
\end{bmatrix}
\end{equation*}
which is the $4\times 4$ identity matrix. By Theorem NMRRI the original matrix must be nonsingular.