## Inverse over Z11

Invert the following matrix over $$\mathbb Z_{11}$$.

$$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1\\ 3 & 4 & 5 & 1 & 2\\ 4 & 5 & 1 & 2 & 3\\ 5 & 1 & 2 & 3 & 4 \end{pmatrix}$$

• #### 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: $$\begin{pmatrix} 7 & 5 & 5 & 5 & 3\\ 5 & 5 & 5 & 3 & 7\\ 5 & 5 & 3 & 7 & 5\\ 5 & 3 & 7 & 5 & 5\\ 3 & 7 & 5 & 5 & 5 \end{pmatrix}$$