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Print
Worksheet 5.8.1 Worksheet
Find the inverse of the elementary matrix
\(E_{(-2)21} \in \R^{3 \times 3}\) using the method of augmenting
\(E_{(-2)21}\) by the identity matrix and performing elimination (as in
Example 3.10.10 ). Check that your result is consistent with
Theorem 3.7.7 .
Solution .
Be definition we know \(E_{(-2)21}\) is the identity matrix with the \(0\) in row \(2\) column \(1\) replaced by \((-2)\text{.}\) \(E_{(-2)21}=\left(\begin{array}{rrr} 1\amp 0\amp 0\\ -2 \amp 1 \amp 0 \\ 0\amp 0\amp 1 \end{array} \right)\text{.}\) Then we have,
\begin{align*}
\left(\begin{array}{rrr|rrr} 1\amp 0\amp 0\amp 1\amp 0\amp 0\\ -2 \amp 1 \amp 0 \amp 0 \amp 1 \amp 0 \\ 0\amp 0\amp 1\amp 0\amp 0\amp 1 \end{array} \right) \amp \xrightarrow{2R_1+R_2 \rightarrow R_2} \amp \left(\begin{array}{rrr|rrr} 1\amp 0\amp 0\amp 1\amp 0\amp 0\\ 0 \amp 1 \amp 0 \amp 2 \amp 1 \amp 0 \\ 0\amp 0\amp 1\amp 0\amp 0\amp 1 \end{array} \right)
\end{align*}
Thus we see that
\(E_{(-2)21}^{-1} = E_{(2)21}\) as predicted by
Theorem 3.7.7 .
