Let E1 = [1 0 0 2 1 0 0 0 1], E2 = [1 0 0 0 1 0 0 0 3], and A = [1 1 3 0 -1 2 0 5 6]. Calculate E1A and E2E1A. Describe the products in terms of matrices obtained from A by elementary row operations.

Question

Let [latex]E_{1} = \left [ \begin{matrix} 1 & 0 & 0 \ 2 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] , E_{2} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 3 \end{matrix} ight ] ,[/latex] and [latex]A = \left [ \begin{matrix} 1 & 1 & 3 \ 0 & -1 & 2 \ 0 & 5 & 6 \end{matrix} ight ] [/latex]. Calculate [latex]E_{1}A[/latex] and [latex]E_{2}E_{1}A [/latex]. Describe the products in terms of matrices obtained from A by elementary row operations.

Accepted Answer

By matrix multiplication, we get

[latex]E_{1}A = \left [ \begin{matrix} 1 & 0 & 0 \ 2 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 1 & 3 \ 0 & -1 & 2 \ 0 & 5 & 6 \end{matrix} ight ] = \left [ \begin{matrix} 1 & 1 & 3 \ 2 & 1 & 8 \ 0 & 5 & 6 \end{matrix} ight ][/latex]

Observe that [latex]E_{1}A [/latex] is the matrix obtained from A by performing the row operation [latex]R_{2} + 2R_{1} [/latex]. That is, by performing the row operation associated with [latex]E_{1}[/latex].

[latex]E_{2}E_{1}A = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 3 \end{matrix} ight ] \left [ \begin{matrix} 1 & 1 & 3 \ 2 & 1 & 8 \ 0 & 5 & 6 \end{matrix} ight ] = \left [ \begin{matrix} 1 & 1 & 3 \ 2 & 1 & 8 \ 0 & 15 & 18 \end{matrix} ight] [/latex]

Thus, [latex]E_{2}E_{1}A [/latex] is the matrix obtained from A by first performing the row operation [latex]R_{2} + 2R_{1}[/latex], and then performing the row operation associated with [latex]E_{2}[/latex], namely [latex]3R_{3}[/latex].

Question 3.6.4: Let E1 = [1 0 0 2 1 0 0 0 1], E2 = [1 0 0 0 1 0 0 0 3], and ...

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