Equivalent Matrices may not be Similar

Theorem
If two square matrices of order $n > 1$ over a ring with unity $R$ are equivalent, they are not necessarily similar.

Proof
Proof by Counterexample:

Let $\mathbf A = \mathbf I_n$ be the unit matrix of order $n > 1$.

Let $\mathbf B$ be an arbitrary invertible matrix over $R$ of order $n$ that is different from the unit matrix.

Then:


 * $\mathbf I_n^{-1} \mathbf A \mathbf B = \mathbf I_n^{-1} \mathbf I_n \mathbf B = \mathbf B$

showing that $\mathbf A$ and $\mathbf B$ are equivalent.

Let $\mathbf P$ be an invertible square matrix of order $n$.

Then:


 * $\mathbf P^{-1} \mathbf A \mathbf P = \mathbf P^{-1} \mathbf P = \mathbf I_n \ne \mathbf B$

Hence, $\mathbf A$ is not similar to $\mathbf B$.