1.5.9 Systems of algebraic equations

Problem: For the given matrix equation a) solve it by the Gauss method; b) make a substitution check; c) solving (by the Gauss method) the equation \( A X=E \), find \( A^{-1} \); d) check the correctness of the answer by calculation \( A^{-1} A \); e) solve the given equation again using \( A^{-1} \), compare the results. \[ \begin{aligned} A & =\left(\begin{array}{ccccc} 1 & 4 & -2 & 2 & -1 \\ 2 & 8 & -3 & 2 & -2 \\ 2 & 9 & -4 & 2 & -2 \\ 0 & -4 & 2 & -1 & 1 \\ -1 & -1 & 2 & -1 & 1 \end{array}\right), \\ B & =\left(\begin{array}{ccccc} 3 & -1 & 4 & 0 & 1 \\ 5 & 8 & -1 & 4 & 2 \\ 1 & -2 & 4 & 0 & 1 \end{array}\right), \quad A X=B . \end{aligned} \]