The Determinant of a Matrix of Matrices

  • Thread starter EngWiPy
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Hi,

Suppose we have the following matrix:

[tex]\begin{center}\begin{pmatrix}\mathbf{L}&\mathbf{A}^T\\\mathbf{A}&\mathbf{0}\end{pmatrix}\end{center}[/tex]

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
 
38
0
det(-AA')
 
1,366
61
38
0
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.
 
1,366
61

AlephZero

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

Counterexample:

[tex]L = \begin{pmatrix} 2 & 0 \cr 0 & 2 \end{pmatrix} \quad
A = \begin{pmatrix} 1 \cr 0\end{pmatrix}[/tex]

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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