Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m < n$.

Is there any equality or inequality over $|B|$, $|AA^T|$, $|D|$ and $|I|$ or their logarithms.

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m < n$.

Is there any relation between $|B|$, $|AA^T|$, $|D|$ and $|I|$ or their logarithms, in the form of an equality or an inequality?