Properties of a Matrix that diagonalizes another matrix

Question

P = (0 1 2;2 0 3;3 4 0) is a matrix that diagonalizes another matrix A.

Does the matrix (3 4 0;2 0 3;0 1 2) diagonlize A? When i swap the top row and bottom row with each other, Will this matrix still be able to diagonalize A?

Answer 1

In general, no. Asserting that $P$ diagonalizes $A$, that is, asserting that $P^{-1}AP$ is a diagonal matrix is equivalent to asserting that the columns of $P$ are eigenvectors of $A$ (and that $P$ is invertible). In general, if you swap the rows like that, you will not have eigenvectors anymore.