Answer
Inverse is not defined.
Work Step by Step
Set up a matrix $[B|I]$, and using row operations, transform into $[I|B^{-1}].$
$\left[\begin{array}{rrrr|rrrr}
1 & 0 & 1 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\
1 & 1 & 1 & 0 & 0 & 0 & 1 & 0\\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 1
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
-R_{1}.\\
-R_{1}.
\end{array}\right. $
$\left[\begin{array}{rrrr|rrrr}
1 & 0 & 1 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\
0 & 1 & 0 & 0 & -1 & 0 & 1 & 0\\
0 & 1 & 0 & 1 & -1 & 0 & 0 & 1
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
-R_{2}.\\
-R_{2}.
\end{array}\right. $
$\left[\begin{array}{rrrr|rrrr}
1 & 0 & 1 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & \fbox{$0$} & -1 & -1 & -1 & 1 & 0\\
0 & 0 & 0 & 0 & -1 & -1 & 0 & 1
\end{array}\right]$
There is a row of zeros on the LHS matrix,
there is no way to bring a $1 $ to the pivot position (3,3),
so the matrix $I_{4}$ can not be obtained on the LHS...
Inverse is not defined.
