Theorem NMRRI tells us we can answer this question by simply row-reducing the matrix. Doing this we obtain,
\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*}
Since the reduced row-echelon form of the matrix is the $4\times 4$ identity matrix $I_4$, we know that $B$ is nonsingular.