1.5.36 Systems of algebraic equations
Condition: Given a homogeneous system of linear equations \( \left\{\begin{array}{c}x_{1}+3 x_{2}+x_{3}-4 x_{4}=0 \\ 2 x_{1}+x_{2}+2 x_{3}-3 x_{4}=0 \\ 3 x_{1}+4 x_{2}+3 x_{3}-7 x_{4}=0\end{array}\right. \) a) find the rank of the system matrix; b) show that the system has a non-trivial solution; c) solve the system: write down the general solution, find a non-trivial particular solution and check it.
Solving systems of algebraic equations by the methods of Gauss, Jordan-Gauss, Cramer and using the inverse matrix. Homogeneous and non-homogeneous systems of equations.