# How can we solve this question without brute force

If $A\in GL_n(R)$, where $R$ is a commutative ring with identity, I would like to prove

$$M=\begin{pmatrix} A & 0 \\ 0 & A^{-1} \\ \end{pmatrix}\in E_{2n}(R)$$

i.e., $M$ is a product of elementary matrices, by elementary matrices I mean just the operations on the identity matrix $I$ defined by $L_j\mapsto \lambda L_i+L_j$, where $L_j$ is a colunm or a row.

I'm trying to solve this question by brute force, making these operations on this matrix and showing this matrix is equivalent to the identity one. I would like to know if there is a simpler and more elegant method to solve this question.

I need help

Since $M$ is a block matrix with "off-diagonal" entries zero, we have $$\det (M) = \det (A) \det(A^{-1}) = 1 \neq 0.$$ Since $1$ is a unit in $R$, we have that $M$ is invertible. Since $M$ is invertible it is a product of elementary matrices.