Inverse over Z11
Task number: 2486
Invert the following matrix over \(\mathbb Z_{11}\).
\( \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1\\ 3 & 4 & 5 & 1 & 2\\ 4 & 5 & 1 & 2 & 3\\ 5 & 1 & 2 & 3 & 4 \end{pmatrix} \)
Hint
Add all rows first.
Resolution
Add all rows to the last one and multiply by \(4^{-1}=3\)
\( \left( \begin{array}{ccccc|ccccc} 1 & 2 & 3 & 4 & 5 & 1 & 0 & 0 & 0 & 0\\ 2 & 3 & 4 & 5 & 1 & 0 & 1 & 0 & 0 & 0\\ 3 & 4 & 5 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\ 4 & 5 & 1 & 2 & 3 & 0 & 0 & 0 & 1 & 0\\ 5 & 1 & 2 & 3 & 4 & 0 & 0 & 0 & 0 & 1 \end{array} \right) \sim \left( \begin{array}{ccccc|ccccc} 1 & 2 & 3 & 4 & 5 & 1 & 0 & 0 & 0 & 0\\ 2 & 3 & 4 & 5 & 1 & 0 & 1 & 0 & 0 & 0\\ 3 & 4 & 5 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\ 4 & 5 & 1 & 2 & 3 & 0 & 0 & 0 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 3 & 3 & 3 & 3 & 3 \end{array} \right) \)
By the last row eliminate the first column
\( \left( \begin{array}{ccccc|ccccc} 1 & 2 & 3 & 4 & 5 & 1 & 0 & 0 & 0 & 0\\ 2 & 3 & 4 & 5 & 1 & 0 & 1 & 0 & 0 & 0\\ 3 & 4 & 5 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\ 4 & 5 & 1 & 2 & 3 & 0 & 0 & 0 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 3 & 3 & 3 & 3 & 3 \end{array} \right) \sim \left( \begin{array}{ccccc|ccccc} 0 & 1 & 2 & 3 & 4 & 9 & 8 & 8 & 8 & 8\\ 0 & 1 & 2 & 3 &10 & 5 & 6 & 5 & 5 & 5\\ 0 & 1 & 2 & 9 &10 & 2 & 2 & 3 & 2 & 2\\ 0 & 1 & 8 & 9 &10 &10 &10 &10 & 0 &10\\ 1 & 1 & 1 & 1 & 1 & 3 & 3 & 3 & 3 & 3 \end{array} \right) \)
We subtract the three pairs of consecutive rows.
\( \left( \begin{array}{ccccc|ccccc} 0 & 1 & 2 & 3 & 4 & 9 & 8 & 8 & 8 & 8\\ 0 & 1 & 2 & 3 &10 & 5 & 6 & 5 & 5 & 5\\ 0 & 1 & 2 & 9 &10 & 2 & 2 & 3 & 2 & 2\\ 0 & 1 & 8 & 9 &10 &10 &10 &10 & 0 &10\\ 1 & 1 & 1 & 1 & 1 & 3 & 3 & 3 & 3 & 3 \end{array} \right) \sim \left( \begin{array}{ccccc|ccccc} 0 & 1 & 2 & 3 & 4 & 9 & 8 & 8 & 8 & 8\\ 0 & 0 & 0 & 0 & 5 & 7 & 9 & 8 & 8 & 8\\ 0 & 0 & 0 & 5 & 0 & 8 & 7 & 9 & 8 & 8\\ 0 & 0 & 5 & 0 & 0 & 8 & 8 & 7 & 9 & 8\\ 1 & 1 & 1 & 1 & 1 & 3 & 3 & 3 & 3 & 3 \end{array} \right) \)
The rest is straightforward.
Result
The inverse matrix is: \( \begin{pmatrix} 7 & 5 & 5 & 5 & 3\\ 5 & 5 & 5 & 3 & 7\\ 5 & 5 & 3 & 7 & 5\\ 5 & 3 & 7 & 5 & 5\\ 3 & 7 & 5 & 5 & 5 \end{pmatrix} \)
