Subsection DROEM Determinants, Row Operations, Elementary Matrices

As a final preparation for our two most important theorems about determinants, we prove a handful of facts about the interplay of row operations and matrix multiplication with elementary matrices with regard to the determinant. But first, a simple, but crucial, fact about the identity matrix.

Theorem DIM Determinant of the Identity Matrix

For every $n\geq 1$, $\detname{I_n}=1$.

Proof

Theorem DEM Determinants of Elementary Matrices

For the three possible versions of an elementary matrix (Definition ELEM) we have the determinants,

$\detname{\elemswap{i}{j}}=-1$
$\detname{\elemmult{\alpha}{i}}=\alpha$
$\detname{\elemadd{\alpha}{i}{j}}=1$

Proof

Theorem DEMMM Determinants, Elementary Matrices, Matrix Multiplication

Suppose that $A$ is a square matrix of size $n$ and $E$ is any elementary matrix of size $n$. Then \begin{equation*} \detname{EA}=\detname{E}\detname{A} \end{equation*}

Proof