Consider the following block matrix

$A=\pmatrix{A_1 & A_2\cr kA_2^\top & A_3}$

where $A_1$ is a symmetric matrix, $A_3$ is diagonal matrix and all entries of $A$ are real and non-negative. How can we show that all eigenvalues of $A$ are real?

Note: $A_2$ is not a square matrix.

Thanks in advance