# Eigenvalue of 'extended block' matrix

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?

-

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 $-63$.