The Determinant of a Matrix of Matrices

[EngWiPy]

Hi,

Suppose we have the following matrix:

$$\begin{center}\begin{pmatrix}\mathbf{L}&\mathbf{A}^T\\\mathbf{A}&\mathbf{0}\end{pmatrix}\end{center}$$

where L is n-by-n matrix, A is m-by-n matrix. How to find the determinant of this square matrix?

Thanks in advance

 

[Some Pig]

det(-AA')

 

[EngWiPy]

det(-AA')

Thank you for replying, but can you elaborate more, please?

 

[Some Pig]

Terms containing elements of L will contains zeroes,
so terms only containing elements of A and A'.
The negative sign indicates orders of the elements.

 

[EngWiPy]

Terms containing elements of L will contains zeroes...

Why is that?

 

[AlephZero]

det(-AA')

This is wrong.

Counterexample:

$$L = \begin{pmatrix} 2 & 0 \cr 0 & 2 \end{pmatrix} \quad <br /> A = \begin{pmatrix} 1 \cr 0\end{pmatrix}$$

Working out the 3x3 determinant shows the mistake in the "proof" that it was right. The only non-zero product in the determinant does contain an element of L.

I don't think there is any "simple" formula for this.