# 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?

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

Homework Helper
det(-AA')
This is wrong.

Counterexample:

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

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.

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