Finding determinant of following matrix I need to find determinant of following matrix. 

$$\begin{bmatrix}1&amp;0&amp;0&amp;0&amp;0&amp;2\\0&amp;1&amp;0&amp;2&amp;2&amp;0\\0&amp;0&amp;1&amp;2&amp;0&amp;0\\0&amp;0&amp;2&amp;1&amp;0&amp;0\\0&amp;2&amp;0&amp;0&amp;1&amp;0\\2&amp;0&amp;0&amp;0&amp;0&amp;1\end{bmatrix}$$

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I did it by simply doing $R_5$ - $R_1$. and then evaluating the determinant. But its a lengthy process but answer came out..

But another thing i have noticed afterwards is that it is symmetric matrix. So finding determinant my be easy, but i don't know how it helps which is my main concern. Thanks
Might be easiest to row-reduce just enough so that the matrix becomes a triangular matrix. Then the determinant is the product of the diagonal entries. In this case, we can row reduce to get $$\begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; 0 &amp;0 &amp; 2\\0&amp; 1&amp;0&amp; 2 &amp; 2 &amp;0\\0 &amp; 0 &amp; 1&amp;2&amp;0&amp;0\\0&amp;0&amp;0&amp;-3&amp;0&amp;0\\0&amp;0&amp;0&amp;0&amp;-3&amp;0\\0&amp;0&amp;0&amp;0&amp;0&amp;-3\end{bmatrix},$$ and so the determinant is $-27$.
Why not $\;R_5-2R_1\;$ ? Adding scalar multiples of one row to other one doesn't change the determinant. Then you can develop wrt the first column:

$$\begin{vmatrix}
1&amp;0&amp;0&amp;0&amp;0&amp;2\\
0&amp;1&amp;0&amp;2&amp;2&amp;0\\
0&amp;0&amp;1&amp;2&amp;0&amp;0\\
0&amp;0&amp;2&amp;1&amp;0&amp;0\\
0&amp;2&amp;0&amp;0&amp;1&amp;0\\
2&amp;0&amp;0&amp;0&amp;0&amp;1\end{vmatrix}\stackrel{R_5-2R_1}\longrightarrow\begin{vmatrix}
1&amp;0&amp;0&amp;0&amp;0&amp;2\\
0&amp;1&amp;0&amp;2&amp;2&amp;0\\
0&amp;0&amp;1&amp;2&amp;0&amp;0\\
0&amp;0&amp;2&amp;1&amp;0&amp;0\\
0&amp;2&amp;0&amp;0&amp;1&amp;0\\
0&amp;0&amp;0&amp;0&amp;0&amp;\!\!-3\end{vmatrix}=1\begin{vmatrix}
1&amp;0&amp;2&amp;2&amp;0\\
0&amp;1&amp;2&amp;0&amp;0\\
0&amp;2&amp;1&amp;0&amp;0\\
2&amp;0&amp;0&amp;1&amp;0\\
0&amp;0&amp;0&amp;0&amp;\!\!-3\end{vmatrix}=$$

$$=-3\begin{vmatrix}
1&amp;0&amp;2&amp;2\\
0&amp;1&amp;2&amp;0\\
0&amp;2&amp;1&amp;0\\
2&amp;0&amp;0&amp;1\\\end{vmatrix}=\ldots etc.$$