Determinant of product of matrices

LagrangeEuler · 2013-02-27T00:23:15+00:00

Do you know where can I find proven identity det(AB)=det(A)det(B) using Levi Civita symbol.

Thread starter LagrangeEuler
Start date Feb 26, 2013
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Determinant Matrices Product

In summary, the determinant of a product of matrices is equal to the product of the determinants of the individual matrices. To calculate the determinant, we find the determinants of the individual matrices and multiply them together. The determinant can be negative, and its sign is determined by the number of row swaps required to reduce the matrix to its reduced row echelon form. It tells us the scaling factor and linear independence of the transformation represented by the matrices, and it also affects the solution of a system of linear equations. If the determinant is non-zero, the system has a unique solution, but if it is zero, there may be no solution or an infinite number of solutions depending on the consistency of the equations.

Feb 26, 2013

LagrangeEuler

Do you know where can I find proven identity
[tex]det(AB)=det(A)det(B)[/tex]
using Levi Civita symbol.

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Feb 26, 2013

micromass

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You can find a proof in "Linear Algebra" by Lang. He doesn't exactly use the Levi-Civita symbol, but he rathers works with signs of permutations. But this is of course almost the same as the Levi-Civita symbol, so it should be very easy to rewrite the proof.

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

The determinant of a product of matrices is equal to the product of the determinants of the individual matrices. This means that if we have matrices A and B, the determinant of their product AB is equal to the determinant of A multiplied by the determinant of B.

2. How do you calculate the determinant of a product of matrices?

To calculate the determinant of a product of matrices, we first find the determinants of the individual matrices. Then, we multiply these determinants together to get the determinant of the product. This can be expressed as det(AB) = det(A) * det(B).

3. Can the determinant of a product of matrices be negative?

Yes, the determinant of a product of matrices can be negative. The sign of the determinant is determined by the number of row swaps required to reduce the matrix to its reduced row echelon form. If an odd number of row swaps is required, the determinant will be negative.

4. What does the determinant of a product of matrices tell us?

The determinant of a product of matrices tells us the scaling factor of the transformation represented by the matrices. It also gives us information about the linear independence of the vectors in the matrices and the orientation of the resulting shape.

5. How does the determinant of a product of matrices affect the solution of a system of linear equations?

The determinant of a product of matrices plays a crucial role in finding the solution to a system of linear equations. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system either has no solution or an infinite number of solutions depending on the consistency of the equations.