Find the 2 × 2 elementary matrix corresponding to each row operation by performing the row operation on the 2 × 2 identity matrix. (a) R2- 3R1 (b) 3R2 (c) R1 ↑↓ R2

Question

Find the 2 × 2 elementary matrix corresponding to each row operation by performing the row operation on the 2 × 2 identity matrix.

[latex](a) R_{2} - 3R_{1} \ \ \ \ \ \ \ \ (b) 3R_{2} \ \ \ \ \ \ \ \ (c) R_{1} \updownarrow R_{2}[/latex]

Accepted Answer

For (a) we have

[latex]\left [ \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} ight ] \begin{matrix} \ R_{2} - 3R_{1} \end{matrix} \ \ \ \ \sim \left [ \begin{matrix} 1 & 0 \ -3 & 1 \end{matrix} ight ] [/latex]

Thus, the corresponding elementary matrix is [latex]E_{1} = \left [ \begin{matrix} 1 & 0 \ -3 & 1 \end{matrix} ight ] [/latex].

For (b) we have

[latex]\left [ \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} ight ] \begin{matrix} \ 3R_{2} \end{matrix} \ \ \ \ \sim \left [ \begin{matrix} 1 & 0 \ 0 & 3 \end{matrix} ight ] [/latex]

Thus, the corresponding elementary matrix is [latex]E_{2} = \left [ \begin{matrix} 1 & 0 \ 0 & 3 \end{matrix} ight ] [/latex].

For (c) we have

[latex]\left [ \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} ight ] \begin{matrix} \ R_{1} \updownarrow R_{2} \end{matrix} \ \ \ \ \sim \left [ \begin{matrix} 0 & 1 \ 1 & 0 \end{matrix} ight ][/latex]

Thus, the corresponding elementary matrix is [latex]E_{3} = \left [ \begin{matrix} 0 & 1 \ 1 & 0 \end{matrix} ight ] [/latex].

Question 3.6.1: Find the 2 × 2 elementary matrix corresponding to each row o...

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