Elementary matrix

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Section 2.4 Elementary matrix

Elementary matrices is very useful to rewrite the elemetary row operations. It is important because we find an algebraic way to describe the row operations. The algebraic way can be understood by computer!

Elementary matrices.

From the identity matrix $I_n\text{,}$ the matrix performed one and only one elementary row operation is called an elementary matrix.

Theorem 2.4.1.

Let$A$ be an $m\times n$ matrix. Let $E$ be the elementary matrix obtained by performing an elementary row(column) operation on $I_{m}$($I_{n}$). If that same elementary row(column) operation is performed on $A\text{,}$ then the resulting matrix is the product $E A(AE)\text{.}$

Examples:

Theorem 2.4.2.

Let $A$ be an $m\times n$ matrix and $B$ be the reduced echelon form of $A\text{.}$ Then there exists elementary matrices $E_1, E_2,\ldots, E_n$ such that

\begin{equation*} B=E_n\ldots E_2E_1A \end{equation*}

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