Similar Eigenvalues of Invertible Matrices

[muzziMsyed21]

Homework Statement

Let A and C be nxn matrices with C invertible. Prove that A and C^-1AC have the same eigenvalues.

Homework Equations

B=C^-1AC

The Attempt at a Solution

det(A-λI) =det(B-λI)
det(A-λI) =det(C^-1AC - λI)
det(A-λI) =det(C^-1AC - λC^-1IC)
det(A-λI) =det[CC^-1(A-λI)] <<<< can you factor out a CC^-1??

 

[Dick]

Homework Statement

Let A and C be nxn matrices with C invertible. Prove that A and C^-1AC have the same eigenvalues.

Homework Equations

B=C^-1AC

The Attempt at a Solution

det(A-λI) =det(B-λI)
det(A-λI) =det(C^-1AC - λI)
det(A-λI) =det(C^-1AC - λC^-1IC)
det(A-λI) =det[CC^-1(A-λI)] <<<< can you factor out a CC^-1??

You can't say C^(-1)AC=C^(-1)CA. The matrices A and C might not commute. Back up a step and think about it again.