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

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

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