1.5.13 Systems of algebraic equations

Problem: Given the system of three linear equations with three unknowns. Required: 1) to find its solution using Cramer's formulas; 2) to solve by the Gauss method; 3 ) to write the system in matrix form and solve it using matrix calculus. Check the correctness of the calculation of the inverse matrix using matrix multiplication. \[ \left\{\begin{array}{c} -3 x_{1}+5 x_{2}-6 x_{3}=-5 \\ 2 x_{1}-3 x_{2}+5 x_{3}=8 \\ x_{1}+4 x_{2}-x_{3}=1 \end{array}\right. \]