elementary matrix
Elementary matrix
An elementary matrix is a matrix obtained from a single elementary row operation to the identity matrix.
Examples of elementary matrices are \[\begin{pmatrix} 0&1\\1&0 \end{pmatrix}\], which is obtained by doing \[r_{1}\leftrightarrow r_{2}\] to \[I_{2}\], and \[\begin{pmatrix} 1&2&0\\0&1&0\\0&0&1 \end{pmatrix}\], which is obtained by doing \[r_{1}+2r_{2}\to r_{1}\].
An elementary row operation on a matrix has the same effect as multiplying by an elementary matrix
Doing a row operation \[r\] to a matrix has the same effect as multiplying that matrix on the left by the elementary matrix corresponding to \[r\].
To demonstrate this, \[A=\begin{pmatrix} 1&3&0&7\\ 8&2&3&11\\ 0&1&5&9\\ \end{pmatrix}\], performing \[r_{2}+3r_{3}\to r_{2}\] will yield us \[A^{\prime}=\begin{pmatrix} 1&3&0&7\\ 8&5&18&38\\ 0&1&5&9 \end{pmatrix}\]. We can achieve the same results by doing \[\begin{pmatrix} 1&0&0\\0&1&3\\0&0&1 \end{pmatrix}\begin{pmatrix} 1&3&0&7\\ 8&2&3&11\\ 0&1&5&9\\ \end{pmatrix}=\begin{pmatrix} 1&3&0&7\\ 8&5&18&38\\ 0&1&5&9\\ \end{pmatrix}\].
Theorem: Let \[r\] be a row operation and \[A\] an \[m\times n\] matrix. Then \[r(A)=r(I_{m})A\].
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