Eigenvalue of block matrix

Question

Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of
A I
I A

where I is the 4 × 4 identity matrix?

(A) -5 (B) -7 (C) 2 (D) 1

I am not able to find out relation between eigenvalues of matrix A and of the given block matrix.

Answer 1

The block matrix can be written as:

$$C=A \otimes I_2 + I_4 \otimes J_2$$

where:

$$J_2=\left (\begin{array}{cc} 0 &\ 1\\ 1 &\ 0 \end{array} \right )$$

Denote the eigenvectors of $A$ by:

$Av_{\lambda} = \lambda v_{\lambda}$

and $J_2$ by:

$J_2w_{\pm} = \pm w_{\pm}$

(The eigenvalues of $J_2$ are $1$ and $-1$). Clearly, the eight combinations:

$v_{\lambda} \otimes w_{\pm}$

are all eigenvectors of $C$.

Thus the eigenvalues of C are $\lambda \pm 1$

Thus the right answer is C corresponding to $\lambda = 1$

Answer 2

Show that if $\alpha$ is an eigenvalue of $A$ and $\beta$ is an eigenvalue of $$B=\begin{pmatrix} \alpha&1\\ 1&\alpha \end{pmatrix}$$ then $\beta$ is also an eigenvalue of $$C=\begin{pmatrix} A&I\\ I&A \end{pmatrix}.$$ (you should be able to write down an eigenvector for $C$in terms of ones for $A$ and for $B$).