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Determinant of block matrices

Given a real n \times n matrix A show that \begin{vmatrix} A &A^2 \\ A^3 & A^4 \end{vmatrix} =0.

Solution

We simply note that

    \[\begin{pmatrix} A &\mathbb{O} \\ A^3 & \mathbb{O} \end{pmatrix} \cdot \begin{pmatrix} \mathbb{I} & A\\ \mathbb{O} & \mathbb{O} \end{pmatrix} = \begin{pmatrix} A &A^2 \\ A^3 & A^4 \end{pmatrix}\]

The rest is … history!

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