Write the matrix $$\begin{bmatrix} 1 & 2 \\\ 3 & 4 \end{bmatrix}$$ as a product of elementary matrices, using as few as you can, and prove that your expression is as short as possible.
I wrote the matrix as a product of elementary matrices
$$\begin{bmatrix} 1 & 2 \\\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\\ 3 & 1 \end{bmatrix}\begin{bmatrix} 1 & -1 \\\ 0 &1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\\ 0 & -2 \end{bmatrix}$$
However, I am facing problem to prove that this expression is the shortest one .
Can I say we will require at least 3 row reduction to convert $$\begin{bmatrix} 1 & 2 \\\ 3 & 4\end{bmatrix}$$ into a identity matrix. Is this a valid proof.