1.5.12 Systems of algebraic equations

Problem: For the given matrix equation a. Solve it using 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 calculating \( A^{-1} A \) : e. Solve the given equation again using \( A^{-1} \), compare the results. \[ \begin{array}{l} A=\left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 83 & -47 & 1 & 0 & 0 \\ -55 & 94 & 0 & 1 & 0 \\ 62 & -71 & 0 & 0 & 1 \end{array}\right), \\ B=\left(\begin{array}{ccccc} 2 & -3 & 4 & -1 & 0 \\ 4 & -2 & 1 & 0 & 3 \end{array}\right), \quad X A=B . \end{array} \]