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A nonsingular matrix can be written as a product of elementary matrices.

Created over 8 years ago, updated 10 days ago

Theorem. Every invertible (nonsingular) matrix $A$ can be written as a product of elementary matrices:
$$A = E_k E_{k-1} \cdots E_2 E_1$$
where each $E_i$ is an elementary matrix.

Proof sketch. If $A$ is nonsingular, it can be reduced to the identity matrix by a finite sequence of elementary row operations. If $E_1, E_2, \ldots, E_k$ are the corresponding elementary matrices, then:
$$E_k E_{k-1} \cdots E_1 A = I$$
which gives $A = E_1^{-1} E_2^{-1} \cdots E_k^{-1}$. Since the inverse of an elementary matrix is also elementary, the result follows.

Example. The matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ can be reduced to $I$ by row operations, so $A = E_1^{-1} E_2^{-1} E_3^{-1}$ for appropriate elementary matrices.