Are matrices $A$ and $B$ similar?

Question

Let $$A = \begin{bmatrix} a & b\\ c & d \end{bmatrix}$$ and let $B$ be the matrix you get by permuting the first and second columns of $A$, i.e., $$B = \begin{bmatrix} b & a\\ d & c \end{bmatrix}$$ Are matrices $A$ and $B$ similar?

I know that by multiplying $A$ by the elementary matrix for column permutation, you get

$$A \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} = B$$

but I want to know if you can find $P \in \mbox{GM}_2(\mathbb R)$ such that $B = P^{-1}AP$.

Answer 1

They are almost never similar. In fact, if $\det A\neq0$, then, since $\det B=-\det A$, they are not similar.