Linear Algebra by Gilbert strang, 4th edition, 5.1.13

Question

If B has eigenvalues 1, 2, 3, C has eigenvalues 4, 5, 6, and D has eigenvalues 7, 8,
9, what are the eigenvalues of the 6 by 6 matrix A =[ B C , 0 D] ?

Answer 1

$det\left \{ \begin{pmatrix} B & C \\ 0 & D \end{pmatrix}-\lambda I \right \} = 0$ This must hold.

Now, (B-$\lambda$)(D-$\lambda$) = 0

this will give eigen values of A which is eigen values of B and D (i.e 1, 2, 3, 7, 8,9)