# Matrix determinants and inverses

#### bakin

1. The problem statement, all variables and given/known data
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?]

2. Relevant equations

3. 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?

#### Dick

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

#### HallsofIvy

1. The problem statement, all variables and given/known data
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?]

2. Relevant equations

3. 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).

#### bakin

Is it:

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

=det(In)det(A)

=1det(A)

=det(A)

????

#### Dick

Homework Helper
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 seperate 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

Homework Helper
So first put them in 3 seperate 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?

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving