How to I prove this fact about two sided ideals?

Suppose $I$ is a two-sided ideal of $M_{2\times2}(\mathbb Q)$. Let $0\neq A\in I$. We consider two cases:

Case 1: $\operatorname{rank}(A)=2$. Then we can perform row/column operations to $A$ to get $\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)\in I$. Therefore $I=M_{2\times2}(\mathbb Q)$.

Case 2: $\operatorname{rank}(A)=1$. Then we can perform row/column operations to $A$ to get $\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right)\in I$ and $\left(\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right)\in I$. Therefore $\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right)+\left(\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right)\in I$, and hence, $I=M_{2\times2}(\mathbb Q)$.