Let A = [1 0 1 1 0 3 -1 1 -1]. Write A and A^-1 as products of elementary matrices.

Question

Let [latex]A = \left [ \begin{matrix} 1 & 0 & 1 \ 1 & 0 & 3 \ -1 & 1 & -1 \end{matrix} ight ][/latex]. Write A and [latex]A^{-1} [/latex] as products of elementary matrices.

Accepted Answer

Row reducing A to RREF gives

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

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

Hence,

[latex]E_{1} = \left [ \begin{matrix} 1 & 0 & 0 \ -1 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] E_{2} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{matrix} ight ] E_{3} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1/2 & 0 \ 0 & 0 & 1 \end{matrix} ight ] E_{4} = \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{matrix} ight ] E_{5} = \left [ \begin{matrix} 1 & 0 & -1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] [/latex]

and [latex]E_{5} E_{4} E_{3} E_{2} E_{1} A = I.[/latex] Therefore,

[latex]A^{-1} = E_{5} E_{4} E_{3} E_{2} E_{1} = \left [ \begin{matrix} 1 & 0 & -1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1/2 & 0 \ 0 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 1 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ -1 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] [/latex]

and

[latex]A = E_{1} ^{-1} E_{2}^{-1} E_{3}^{-1} E_{4}^{-1} E_{5}^{-1} = \left [ \begin{matrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ -1 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 1 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{matrix} ight ] \left [ \begin{matrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{matrix} ight ] [/latex]

Question 3.6.6: Let A = [1 0 1 1 0 3 -1 1 -1]. Write A and A^-1 as products ...

Related Answered Questions