<s>[INST] Computing the determinant of a large matrix? How would I go about computing the determinant of large matrices, such as $6 \times 6$. I believe that I need to use multilinear maps, but I am not sure how I can go about computing the determinant in a nice and efficient way. 
Can anyone show me how I can determine the determinant of the matrix below in a simple and efficient way?
\begin{pmatrix}
0 &  0&  1&  1& 1 & 1\\ 
1 & 0 & 0 &  0&  0& 1\\ 
1 & 0 &  1& 1 & 1 &1 \\ 
0 & 1 & 1 & 1 & 0 &1 \\ 
0 & 1 &  0& 1 &  0& 0\\ 
0 &  0&  1& 0 &  0& 0
\end{pmatrix}
 [/INST] You can use Laplace's expansion of the  determinant. 
More precisely, begin expanding by the last row, then some row manipulation and expanding by convenient rows/columns ends up in a $2\times2$ determinant:
\begin{align}
\begin{vmatrix}
0&0&1&1&1&1 \\ 1&0&0&0&0&1 \\ 1&0&1&1&1&1 \\
0&1&1&1&0&1 \\ 0&1&0&1&0&0 \\ 0&0&\color{red}1&0&0&0\end{vmatrix}&=
-\begin{vmatrix}
0&0&1&1&1 \\ 1&0&0&0&1 \\ 1&0&1&1&1 \\
0&1&1&0&1 \\ 0&1&1&0&0\end{vmatrix}=
-\begin{vmatrix}
0&0&1&1&1 \\ 1&0&0&0&1 \\ 1&0&1&1&1 \\
0&0&0&0&\color{red}1 \\ 0&1&1&0&0\end{vmatrix}=
+\begin{vmatrix}
0&0&1&1 \\ \color{red}1&0&0&0 \\ 1&0&1&1 \\ 0&1&1&0 \end{vmatrix}\\
&=-\begin{vmatrix}
0&1&1 \\ 0&1&1 \\ \color{red}1&1&0 \end{vmatrix}=-\begin{vmatrix}
1&1 \\ 1&1  \end{vmatrix}=0.
\end{align}
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