Similar Matrices: Showing A Not Similar to B

[cutesteph]

Homework Statement

A=[a,1;0,a] B=[a,0;0,a]
If I want to show if matrix A is NOT similar to matrix B. Is it enough to show that B=/=Inv(P)*A*P? Or would I need to show that they do not have both the same eigenvalues and corresponding eigenvectors?

 

[Dick]

Homework Statement

A=[a,1;0,a] B=[a,0;0,a]
If I want to show if matrix A is NOT similar to matrix B. Is it enough to show that B=/=Inv(P)*A*P? Or would I need to show that they do not have both the same eigenvalues and corresponding eigenvectors?

It would be enough to show B=/=Inv(P)*A*P for any invertible matrix P, if you have a clever way to do that. But I think the eigenvalue/eigenvector approach is more straightforward. If you calculate those for each matrix what do you get?