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: \( \begin{pmatrix} 7 & 5 & 5 & 5 & 3\\ 5 & 5 & 5 & 3 & 7\\ 5 & 5 & 3 & 7 & 5\\ 5 & 3 & 7 & 5 & 5\\ 3 & 7 & 5 & 5 & 5 \end{pmatrix} \)

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
Cs translation
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