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}

First, note that row 6 is mostly '$0$'s. Next note the red values in row 2. The non-zero part of the determinant has to go through that $1$, so swap rows 2 and 5,
\begin{align}
\begin{vmatrix}
0&0&1&1&1&1 \\ 
1&\color{red}0&0&\color{red}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&1&0&1&0&0 \\ 
1&0&1&1&1&1 \\
0&1&1&1&0&1 \\ 
1&\color{red}0&0&\color{red}0&0&1 \\ 
\color{red}0&\color{red}0&1&\color{red}0&\color{red}0&\color{red}0
\end{vmatrix}= 0
\end{align}
which shows that the determinant is zero.
The column solution is even easier:
\begin{align}
\begin{vmatrix}
0&\color{red}0&1&1&1&1 \\ 
1&\color{red}0&0&0&0&1 \\ 
1&\color{red}0&1&1&1&1 \\
\color{red}0&1&1&1&0&1 \\ 
\color{red}0&1&\color{red}0&1&0&0 \\ 
\color{red}0&0&1&0&0&0
\end{vmatrix}& = 0
\end{align}
This is a medium sized matrix at most - to find the determinant for a actual large matrix ($n>100$), look up RRQR.
