<s>[INST] 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.
 [/INST] Permuting the rows and columns, we obtain a block diagonal matrix.
$$\det \begin{bmatrix} \color{red}{1} & 0 & 0 & 0 & 0 & \color{red}{2}\\ 0 & \color{orange}{1} & 0 & 0 & \color{orange}{2} & 0\\ 0 & 0 & \color{magenta}{1} & \color{magenta}{2} & 0 & 0\\ 0 & 0 & \color{magenta}{2} & \color{magenta}{1} & 0 & 0\\ 0 & \color{orange}{2} & 0 & 0 & \color{orange}{1} & 0\\ \color{red}{2} & 0 & 0 & 0 & 0 & \color{red}{1}\end{bmatrix} = \det \begin{bmatrix} \color{red}{1} & \color{red}{2} & & & & \\ \color{red}{2} & \color{red}{1} & & & & \\ & & \color{orange}{1} & \color{orange}{2} & & \\ & & \color{orange}{2} & \color{orange}{1} & & \\ & & & & \color{magenta}{1} & \color{magenta}{2} \\ & & & & \color{magenta}{2} & \color{magenta}{1} \end{bmatrix} = \left( \det \begin{bmatrix} 1 & 2\\ 2 & 1\end{bmatrix} \right)^3 = (-3)^3 = \color{blue}{-27}$$

matrices block-matrices permutation-matrices determinant
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