Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? 
Let $\operatorname{Mat}(n,\mathbb{K})$ be the ring of $n\times n$ matrices with entries in a field $\mathbb{K}$.   Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to  $\operatorname{Mat}(3,\mathbb{R})$?

I want to know whether there is ring isomorphism between the two.
I cannot seem to find any bijection that will do. Thanks.
An Idea.
Let $f:\text{Mat}(2,\mathbb{R})\to\text{Mat}(3,\mathbb{R})$ be a ring homomorphism.  Then, $\ker f$ is a two-sided ideal of $\text{Mat}(2,\mathbb{R})$.  It is known that $\text{Mat}(2,\mathbb{R})$ is a simple ring.  Therefore, either $\ker f=0$ or $\ker f=\text{Mat}(2,\mathbb{R})$.  In the case $\ker f=\text{Mat}(2,\mathbb{R})$, it follows that $f=0$, so it is not an isomorphism.  If $\ker f=0$, then $f$ is an embedding.  This is where the argument stops.  There are many ways to embed $\text{Mat}(2,\mathbb{R})$ into $\text{Mat}(3,\mathbb{R})$.  But is there an embedding that is a surjection, i.e., it is an isomorphism?
 A: There are no nilpotent elements of order $3$ in $\text{Mat}_{2\times2}(\mathbb{R})$, while there are many of them in $\text{Mat}_{3\times3}(\mathbb{R})$.  Hence, the two cannot be isomorphic as rings.
In general, for two positive integers $m$ and $n$, and for any unital (associative) ring $R$, the rings of matrices $\text{Mat}_{m\times m}(R)$ and $\text{Mat}_{n\times n}(R)$ are isomorphic as rings if and only if $m=n$.  If the assumption that the ring $R$ is unital is dropped, then it is possible that $\text{Mat}_{m\times m}(R)$ and $\text{Mat}_{n\times n}(R)$ are isomorphic even if $m\neq n$.  For example, by taking $R=\bigoplus\limits_{i=1}^\infty\,\mathbb{Z}$ with trivial multiplication (i.e., $x\cdot y=0$ for all $x,y\in R$), we have $$\text{Mat}_{m\times m}(R)\cong R\cong\text{Mat}_{n\times n}(R)$$
for all positive integers $m$ and $n$.
On the other hand, if there are two unital rings $R$ and $S$, then it is possible that $\text{Mat}_{m\times m}(R)$ and $\text{Mat}_{n\times n}(S)$ are isomorphic.  An easy example, is when $S=\text{Mat}_{k\times k}(R)$.  Then,
$$\text{Mat}_{(nk)\times (nk)}(R)\cong \text{Mat}_{n\times n}(S)\,.$$
