How to find the determinant of this $6\times 6$ X-matrix? This question was asked in my quiz and i was unable to solve it, so I am asking it here.


Find the value of determinant of this particular matrix .
$$\begin{pmatrix}1&0&0&0&0&2\\0&1&0&0&2&0\\0&0&1&2&0&0\\0&0&2&1&0&0\\0&2&0&0&1&0\\2&0&0&0&0&1\end{pmatrix}$$


I have no clue on how this kind of matrices can be solved. Can anyone give a general strategy on how to solve matrices whose size are greater that $3\times 3$?
That would be really helpful.

Let ${\rm R}_3$ be the $3 \times 3$ reversal matrix. Hence,
$$\det \left[\begin{array}{ccc|ccc} 1&0&0&0&0&2\\ 0&1&0&0&2&0\\ 0&0&1&2&0&0\\ \hline 0&0&2&1&0&0\\0&2&0&0&1&0\\ 2&0&0&0&0&1\end{array}\right] = \det \begin{bmatrix} {\rm I}_3 & 2{\rm R}_3\\ 2{\rm R}_3 & {\rm I}_3\end{bmatrix} = \det \left( {\rm I}_3 - 4 {\rm R}_3^2 \right) = (-3)^3 = \color{blue}{-27}$$
because ${\rm R}_3^2 = {\rm I}_3$.

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