= Elementary-matrix factorization theorem
{title2=$A\text{ invertible}\iff A=E_1\cdots E_r$}
Every invertible square <matrix> is a product of <elementary matrices>. <Gaussian elimination> reduces $A$ to the <identity matrix> by elementary row operations, giving $F_r\cdots F_1A=I$. Reverse their order and invert them to write $A=F_1^{-1}\cdots F_r^{-1}$. Each inverse is again elementary. Conversely every elementary matrix is invertible, so every such product is invertible. The identity may be represented by the empty product.
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