Matrix Similarity is Equivalence Relation/Proof 2

Proof
Checking in turn each of the criteria for equivalence:

Reflexive
$\mathbf A = \mathbf{I_n}^{-1} \mathbf A \mathbf{I_n}$ trivially, for all order $n$ square matrices $\mathbf A$.

So matrix similarity is reflexive.

Symmetric
Let $\mathbf B = \mathbf P^{-1} \mathbf A \mathbf P$.

As $\mathbf P$ is invertible, we have:

So matrix similarity is symmetric.

Transitive
Let $\mathbf B = \mathbf P_1^{-1} \mathbf A \mathbf P_1$ and $\mathbf C = \mathbf P_2^{-1} \mathbf B \mathbf P_2$.

Then:

By Product of Matrices is Invertible iff Matrices are Invertible, $\paren {\mathbf P_1 \mathbf P_2}$ is invertible.

So matrix similarity is transitive.

So, by definition, matrix similarity is an equivalence relation.