Let $X_{ij}(a) = I_n + a E_{ij}$ be an elementary matrix where $E_{ij}$ is the $n \times n$ matrix that has 1 in position $(i,j)$ and zero everywhere else where $a$ is a scalar in some arbitrary field $\mathbb{F}$. How can I show that $X_{ij}(a) \in SL(n,\mathbb{F})$ is conjugate to a matrix of the form $X_{12}(b)$?
So this question is asking to show that elementary matrices are similar. I read that two matrices are similar if they have the same eigenvalues. I can show that 1 is the only eigenvalue of $X_{ij}(a)$ and $X_{12}(b)$ if $i \neq j$.
Is this the correct way to proceed or is there a more straightforward approach?
I also proved every matrix in $SL(n, \mathbb{F})$ can be written as a product of elementary matrices but I'm not sure how that would help.
Hints. Suppose $i\ne j$ and $a\ne0$. You may try to prove that $X_{ij}(a)$ is similar to $X_{12}(a)$ via a permutation matrix, and $X_{12}(a)$ is similar to $X_{12}(1)$ via a diagonal matrix.