
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.
\[
\begin{aligned}
A & =\left(\begin{array}{ccccc}
1 & 4 & -2 & 2 & -1 \\
2 & 8 & -3 & 2 & -2 \\
2 & 9 & -4 & 2 & -2 \\
0 & -4 & 2 & -1 & 1 \\
-1 & -1 & 2 & -1 & 1
\end{array}\right), \\
B & =\left(\begin{array}{ccccc}
3 & -1 & 4 & 0 & 1 \\
5 & 8 & -1 & 4 & 2 \\
1 & -2 & 4 & 0 & 1
\end{array}\right), \quad A X=B .
\end{aligned}
\]