Let A be a $4 \times 4$ matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix?

The block matrix [A I; I A] has eigenvalues that are eigenvalues of A+I and A-I (for eigenvectors where the top and bottom halves are aligned or anti-aligned). For eigenvalue -5 of A: -5+1=-4 and -5-1=-6. For -2: -2+1=-1, -2-1=-3. For 1: 1+1=2, 1-1=0. For 4: 4+1=5, 4-1=3. Therefore 1 is an eigenvalue (from eigenvalue 0).