# Linear algebra problem

## Homework Statement

Let A and B be two n x n matrices that are related by the equation (P^-1)(A)(P) = B, where P is another n x n matrix. Prove that det(A) = det(B).

## The Attempt at a Solution

I'm thinking the first step might be to come up with general forms of A and B that are related by the above equation? I've been trying to do that and not been successful so far. Any ideas? thanks

Hurkyl
Staff Emeritus
Gold Member
Why not just 'compute' det(B)?

## Homework Statement

Let A and B be two n x n matrices that are related by the equation (P^-1)(A)(P) = B, where P is another n x n matrix. Prove that det(A) = det(B).

## The Attempt at a Solution

I'm thinking the first step might be to come up with general forms of A and B that are related by the above equation? I've been trying to do that and not been successful so far. Any ideas? thanks
Two basic facts you should know (and use them in this exercise):
The equality
det(A.B)=det(A).det(B)
is true for any two nxn matrices.
And we have for any invertible matrix
det(A^-1)= ???
(I think you should be able to guess the result using the definition of the inverse and the above equation.)
That's all you need to know in order to solve this one.