# Determining whether two matrices are similar without calculating eigenvalues

How would you prove that $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $\begin{bmatrix} d & c \\ b & a \end{bmatrix}$ are similar? Their characteristic polynomials are identical, but this isn't sufficient, and without actually calculating the eigenvectors and the dimensions of the eigenspaces it seem we can't use the Jordan form.

In general, how would you determine two matrices are similar without numerical calculations, simply based on such relations between their entries?

Let $P=\begin{bmatrix} 0&1\\1&0\end{bmatrix}$. Note that $P=P^{-1}$.
Consider $P\begin{bmatrix} a&b\\c&d\end{bmatrix}P=\begin{bmatrix}d&c\\b&a\end{bmatrix}.$ Hence, your two matrices are similar.