Determinant of Elementary Row Matrix

Theorem
Let $\mathbf E$ be a product of elementary matrices $\mathbf{E}_1\mathbf{E}_2 \cdots \mathbf{E}_k$.

Let $\mathbf D$ be a matrix.

From Effect of Elementary Row Operations on Determinant and definition of elementary matrix, using the conventional matrix product:
 * $\displaystyle \det \left({\mathbf {ED}}\right) = \alpha \det \left({\mathbf D}\right)$

Then $\alpha = \det \left({\mathbf E}\right)$.

Proof
It is only necessary to prove this for when E is the product of just one elementary matrix because then:

because E is just the unit matrix after having k elementary row operations being performed on it.

Consider $\det \left({\mathbf E_1}\right)$, by Effect of Elementary Row Operations on Determinant it is seen that:
 * $\displaystyle \det \left({\mathbf E_1}\right) = \alpha_1 \det \left({\mathbf I}\right)$

From Determinant of Diagonal Matrix it is easily seen that $\det \left({\mathbf I}\right) = 1$, so
 * $\displaystyle \det \left({\mathbf E_1}\right) = \alpha_1$

Also see

 * Determinant of Matrix Product


 * Effect of Elementary Row Operations on Determinant


 * Definition:Determinant