Matrix determinants and inverses

[bakin]

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=I_n, 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=I_n into det(S^-1AS), so you have det(I_nA)=det(A), but I really don't know. Can somebody get me started so we can work through it?

 

[Dick]

Sure. You should have proved det(AB)=det(A)*det(B). det(I)=1. Start from there.

 

[boaz]

Moreover, I've been told that :
[; det(A^-1)=(det(A))^-1 ;]
http://upload.wikimedia.org/math/f/0/c/f0caaf42398716ab6480982a3462fbca.png
In addition to that, we also know that :
[; det(A\cdot B)=det(A)\cdot det(B)=det(B)\cdot det(A) ;]

 

[HallsofIvy]

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=I_n, 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=I_n into det(S^-1AS), so you have det(I_nA)=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=I_n, how are det(S^-1) and det(S) related?" I presume you know that det(AB)= det(A)det(B).

 

[bakin]

Is it:

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

=det(I_n)det(A)

=1det(A)

=det(A)?

 

[Dick]

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.

 

[bakin]

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) ?

 

[Dick]

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.

 

[bakin]

Thanks for the help

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?