Find the inverse of each of the following elementary matrices. Check your answer by multiplying the matrices together. (a) E1 = [1 2 0 1] (b) E2 [0 0 1 0 1 0 1 0 0] (c) E3 = [1 0 0 0 1 0 0 0 -3]

Question

Find the inverse of each of the following elementary matrices. Check your answer by multiplying the matrices together.

[latex](a) \ \ E_{1} = \left [ \begin{matrix} 1 & 2 \ 0 & 1 \end{matrix} ight ] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (b) \ \ E_{2} = \left [ \begin{matrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{matrix} ight ]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (c) \ \ E_{3} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -3 \end{matrix} ight ][/latex]

Accepted Answer

The inverse matrix [latex]E_{1} ^ {-1}[/latex] is the elementary matrix associated with the row operation required to bring [latex]E_{1}[/latex] back to I. That is, [latex]R_{1} - 2R_{2}[/latex]. Therefore,

[latex]E_{1} ^ {-1} = \left [ \begin{matrix} 1 & -2 \ 0 & 1 \end{matrix} ight ][/latex]

Checking, we get [latex]E_{1} ^ {-1} E_{1} = \left [ \begin{matrix} 1 & -2 \ 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 2 \ 0 & 1 \end{matrix} ight ] = \left [ \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} ight ] [/latex].

The inverse matrix [latex]E_{2} ^ {-1}[/latex] is the elementary matrix associated with the row operation required to bring [latex]E_{2}[/latex] back to I. That is,

[latex]R_{1} \updownarrow R_{3}[/latex]. Therefore,

[latex]E_{2} ^ {-1} = \left [ \begin{matrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{matrix} ight ][/latex]

Checking, we get [latex]E_{2} ^ {-1} E_{2} = \left [ \begin{matrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{matrix} ight ]\left [ \begin{matrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{matrix} ight ] = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ][/latex].

The inverse matrix [latex]E_{3} ^ {-1} [/latex] is the elementary matrix associated with the row operation required to bring [latex]E_{3}[/latex] back to I. That is, [latex]({-1}/{3}) R_{3}[/latex]. Therefore,

[latex]E_{3} ^ {-1} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1/3 \end{matrix} ight ] [/latex]

Checking, we get [latex]E_{3} ^ {-1} E_{3} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1/3 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -3 \end{matrix} ight ] = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] [/latex].

Question 3.6.3: Find the inverse of each of the following elementary matrice...

Related Answered Questions