Matrix determinants and inverses

In summary: There are examples of when doing what you did in post #5 would be wrong, but I can't think of any specific examples off the top of my head. Thanks for asking!
  • #1
bakin
58
0

Homework Statement


If A and S are n x n matrices with S invertible, show that det(S-1AS)=det(A). [HINT: Since S-1S=In, how are det(S-1) and det(S) related?]


Homework Equations





The Attempt at a Solution



Not sure. The only thing I can think of doing is substituting S-1S=In into det(S-1AS), so you have det(InA)=det(A), but I really don't know. Can somebody get me started so we can work through it?
 
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  • #2
Sure. You should have proved det(AB)=det(A)*det(B). det(I)=1. Start from there.
 
  • #4
bakin said:

Homework Statement


If A and S are n x n matrices with S invertible, show that det(S-1AS)=det(A). [HINT: Since S-1S=In, how are det(S-1) and det(S) related?]


Homework Equations





The Attempt at a Solution



Not sure. The only thing I can think of doing is substituting S-1S=In into det(S-1AS), so you have det(InA)=det(A), but I really don't know. Can somebody get me started so we can work through it?

You don't seem to have payed any attention to the hint! "S-1S=In, how are det(S-1) and det(S) related?" I presume you know that det(AB)= det(A)det(B).
 
  • #5
Is it:

det(S-1AS) = det(S-1S)det(A)

=det(In)det(A)

=1det(A)

=det(A)?
 
  • #6
That's true. But don't make it look like you think the matrices commute. S^(-1)AS is NOT necessarily equal to S^(-1)SA. But det(S^(-1)AS)=det(S^(-1))*det(A)*det(S). Now you can rearrange.
 
  • #7
So first put them in 3 separate determinants like you did, then move the S-1 next to S, then put it back together? Go backwards from two determinants multiplying each other to a single determinant, which would be equal to det(S-1S) ?
 
  • #8
bakin said:
So first put them in 3 separate determinants like you did, then move the S-1 next to S, then put it back together? Go backwards from two determinants multiplying each other to a single determinant, which would be equal to det(S-1S) ?

That's it exactly.
 
  • #9
Thanks for the help :smile:

Just on a curious note, are there examples of when doing what I did in post #5 would be wrong, or is it just an incorrect way of solving?
 

What is a matrix determinant?

A matrix determinant is a numerical value that is calculated from the elements of a square matrix. It is denoted by |A| or det(A) and represents the scaling factor of the matrix when used in linear transformations.

How is a matrix determinant calculated?

The determinant of a 2x2 matrix is calculated by taking the product of the elements on the main diagonal and subtracting the product of the elements on the off-diagonal. For larger matrices, the determinant can be calculated using various methods such as row reduction or cofactor expansion.

What is the significance of a matrix determinant?

The determinant of a matrix is important in various fields such as mathematics, physics, and engineering. It is used to determine if a matrix has an inverse, to solve systems of linear equations, and to calculate areas and volumes in geometry.

What is a matrix inverse?

A matrix inverse is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is denoted by A-1 and is used to solve systems of linear equations and to perform division of matrices.

How is a matrix inverse calculated?

The inverse of a 2x2 matrix is calculated by swapping the elements on the main diagonal, changing the sign of the elements on the off-diagonal, and dividing each element by the determinant. For larger matrices, the inverse can be calculated using various methods such as row reduction or the adjugate matrix method.

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