## Matrix equation

### Task number: 2485

Find a matrix \(\mathbf A\), that over \(\mathbb Z_5\) satisfies

\(\mathbf A \begin{pmatrix} 4 & 4 & 0 & 1 \\ 3 & 1 & 2 & 2 \\ 2 & 3 & 1 & 3 \\ 3 & 2 & 3 & 4 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & 2 & 3 \\ 3 & 1 & 2 & 2 \\ 2 & 3 & 1 & 3 \\ 1 & 2 & 3 & 4 \\ \end{pmatrix} \)

#### Hint

From \(\mathbf A \mathbf B = \mathbf C\) express \(\mathbf A\).

#### Resolution

\(\mathbf A = \mathbf C \mathbf B^{-1} = \begin{pmatrix} 1 & 0 & 2 & 3 \\ 3 & 1 & 2 & 2 \\ 2 & 3 & 1 & 3 \\ 1 & 2 & 3 & 4 \\ \end{pmatrix} \begin{pmatrix} 4 & 4 & 0 & 1 \\ 3 & 1 & 2 & 2 \\ 2 & 3 & 1 & 3 \\ 3 & 2 & 3 & 4 \\ \end{pmatrix}^{-1} = \begin{pmatrix} 1 & 0 & 2 & 3 \\ 3 & 1 & 2 & 2 \\ 2 & 3 & 1 & 3 \\ 1 & 2 & 3 & 4 \\ \end{pmatrix} \begin{pmatrix} 4 & 1 & 4 & 3 \\ 4 & 1 & 3 & 0 \\ 3 & 4 & 3 & 0 \\ 4 & 2 & 2 & 3 \\ \end{pmatrix} \)

#### Result

\( \mathbf A= \begin{pmatrix} 2 & 0 & 1 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 2 & 3 & 2 & 0 \\ \end{pmatrix} \)