The Determinant of a Matrix of Matrices

In summary, the conversation is discussing how to find the determinant of a square matrix that has a specific structure, where one part of the matrix only contains elements from matrix A. The initial response suggests that the determinant can be found by multiplying the determinants of two smaller matrices, but the other person provides a counterexample to show that this approach is incorrect. It is stated that there may not be a simple formula for finding the determinant of this type of matrix.
  • #1
EngWiPy
1,368
61
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
 
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  • #2
det(-AA')
 
  • #3
Some Pig said:
det(-AA')

Thank you for replying, but can you elaborate more, please?
 
  • #4
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.
 
  • #5
Some Pig said:
Terms containing elements of L will contains zeroes...

Why is that?
 
  • #6
Some Pig said:
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.
 

1. What is the determinant of a matrix of matrices?

The determinant of a matrix of matrices is a numerical value that can be calculated from the elements of the matrices. It is a measure of how the matrices are related to each other and can provide information about their linear dependence.

2. How is the determinant of a matrix of matrices calculated?

The determinant of a matrix of matrices is calculated by first multiplying the elements of the matrices and then subtracting the product of the elements in the opposite diagonal. This process can be repeated for larger matrices by recursively calculating the determinants of submatrices.

3. What does the determinant of a matrix of matrices tell us?

The determinant of a matrix of matrices can tell us several things, including whether the matrices are linearly independent or dependent, the volume of the parallelepiped formed by the column vectors of the matrices, and the orientation of the transformation represented by the matrices.

4. Can the determinant of a matrix of matrices be negative?

Yes, the determinant of a matrix of matrices can be negative. The sign of the determinant is determined by the orientation of the transformation represented by the matrices. If the determinant is negative, it means that the transformation results in a reflection.

5. What are some real-world applications of the determinant of a matrix of matrices?

The determinant of a matrix of matrices has many applications in mathematics, physics, and engineering. It is used to solve systems of linear equations, calculate areas and volumes, and analyze the stability of systems in physics and engineering. It is also used in computer graphics to determine the orientation and position of objects in 3D space.

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