91Ó°ÊÓ

There are 12 possible \(3 \times 3\) elementary matrices and their inverses: 1. \(E_{12} = \begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&1\\ \end{pmatrix}\), \(E_{12}^{-1} = E_{12}\) 2. \(E_{13} = \begin{pmatrix} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{pmatrix}\), \(E_{13}^{-1} = E_{13}\) 3. \(E_{23} = \begin{pmatrix} 1&0&0\\ 0&0&1\\ 0&1&0\\ \end{pmatrix}\), \(E_{23}^{-1} = E_{23}\) 4. \(E_{1k} = \begin{pmatrix} k&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\), \(E_{1k}^{-1} = \begin{pmatrix} 1/k&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\) 5. \(E_{2k} = \begin{pmatrix} 1&0&0\\ 0&k&0\\ 0&0&1\\ \end{pmatrix}\), \(E_{2k}^{-1} = \begin{pmatrix} 1&0&0\\ 0&1/k&0\\ 0&0&1\\ \end{pmatrix}\) 6. \(E_{3k} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&k\\ \end{pmatrix}\), \(E_{3k}^{-1} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1/k\\ \end{pmatrix}\) 7. \(E_{1(2)} = \begin{pmatrix} 1&0&0\\ k&1&0\\ 0&0&1\\ \end{pmatrix}\), \(E_{1(2)}^{-1} = \begin{pmatrix} 1&0&0\\ -k&1&0\\ 0&0&1\\ \end{pmatrix}\) 8. \(E_{1(3)} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ k&0&1\\ \end{pmatrix}\), \(E_{1(3)}^{-1} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ -k&0&1\\ \end{pmatrix}\) 9. \(E_{2(1)} = \begin{pmatrix} 1&k&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\), \(E_{2(1)}^{-1} = \begin{pmatrix} 1&-k&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\) 10. \(E_{2(3)} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&k&1\\ \end{pmatrix}\), \(E_{2(3)}^{-1} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&-k&1\\ \end{pmatrix}\) 11. \(E_{3(1)} = \begin{pmatrix} 1&0&k\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\), \(E_{3(1)}^{-1} = \begin{pmatrix} 1&0&-k\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\) 12. \(E_{3(2)} = \begin{pmatrix} 1&0&0\\ 0&1&k\\ 0&0&1\\ \end{pmatrix}\), \(E_{3(2)}^{-1} = \begin{pmatrix} 1&0&0\\ 0&1&-k\\ 0&0&1\\ \end{pmatrix}\)