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Swap \(i\)th for \(j\)th row in \(A\) \( R_{ij}A=r_{ij}(I_n)A \) |
example:\( A= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \), \( R_{12}= \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)\( A\xrightarrow{\quad R_{12}\quad} \begin{bmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix} \) |
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multiple \(i\)th in \(A\). \( R_i^{(k)}A=r_i^{(k)}(I_n)A, k\neq 0 \) |
example:\( A= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \), \( R_2^{(3)}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)\( A\xrightarrow{\quad R_2^{(3)}\quad} \begin{bmatrix} 1 & 2 & 3 \\ 12 & 15 & 18 \\ 7 & 8 & 9 \end{bmatrix} \) |
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using multiple k of \(i\)th add to \(j\)th row in \(A\),
\( k\in R \). \( R_{ij}^{(k)}A=r_{ij}^{(k)}(I_n)A \) |
example:\( A= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \), \( R_{12}^{(-4)} = \begin{bmatrix} 1 & 0 & 0 \\ -4 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)\( A\xrightarrow{\quad R_{12}^{(-4)}\quad} \begin{bmatrix} 1 & 2 & 3 \\ 4+(-4) & 5+(-8) & 6+(-12) \\ 7 & 8 & 9 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 7 & 8 & 9 \end{bmatrix} \) |