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Thanks guys for the previous answer, Now suppose if I have a matrix e.g

$M_1 = \begin{pmatrix} \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} & B \\ B' & D \end{pmatrix}$ and $M_2$ as

$M_2 = \begin{pmatrix} \begin{pmatrix} a_{11} & -a_{12} \\ -a_{21} & a_{22} \end{pmatrix} & B \\ B' & D \end{pmatrix}$

How can i prove for this as eig($M_1$) = eig($M_2$), can this be proven?

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This isn't true. Consider the matrix $M_1 = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 3 & 3 \end{pmatrix}$ with determinant $9$ and $M_2 = \begin{pmatrix} 1 & -2 & 3 \\ -2 & 1 & 3 \\ 3 & 3 & 3 \end{pmatrix}$ with determinant $-63$.