30 lines
1.8 KiB
TeX
30 lines
1.8 KiB
TeX
\section{Elementary Matrix Operations and Elementary Matrices}
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\begin{definition}
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\hfill\\
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Let $A$ be an $m \times n$ matrix. Any one of the following three operations on the rows [columns] of $A$ is called an \textbf{elementary row [column] operation}:
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\begin{enumerate}
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\item interchanging any two rows [columns] of $A$;
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\item multiplying any row [column] of $A$ by a nonzero scalar;
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\item adding any scalar multiple of a row [column] of $A$ to another row [column].
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\end{enumerate}
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Any of these three operations are called an \textbf{elementary operation}. Elementary operations are of \textbf{type 1}, \textbf{type 2}, or \textbf{type 3} depending on whether they are obtained by (1), (2), or (3).
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\end{definition}
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\begin{definition}
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\hfill\\
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An $n \times n$ \textbf{elementary matrix} is a matrix obtained by performing an elementary operation on $I_n$. The elementary matrix is said to be of \textbf{type 1}, \textbf{2}, or \textbf{3} according to whether the elementary operation performed on $I_n$ is a type 1, 2, or 3 operation, respectively.
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\end{definition}
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\begin{theorem}
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\hfill\\
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Let $A \in M_{m \times n}(\F)$, and suppose that $B$ is obtained from $A$ by performing an elementary row [column] operation. Then there exists an $m \times m$ [$n \times n$] elementary matrix $E$ such that $B = EA$ [$B = AE]$. In fact, $E$ is obtained from $I_m$ [$I_n]$ by performing the same elementary row [column] operation as that which was performed on $A$ to obtain $B$. Conversely, if $E$ is an elementary $m \times m$ [$n \times n$] matrix, then $EA$ [$AE$] is the matrix obtained from $A$ by performing the same elementary row [column] operation as that which produces $E$ from $I_m$ [$I_n$].
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\end{theorem}
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\begin{theorem}
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\hfill\\
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Elementary matrices are invertible, and the inverse of an elementary matrix is an elementary matrix of the same type.
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\end{theorem}
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