Systems of algebraic equations

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Problem: Solve the matrix equation \[ A X=B, A=\left(\begin{array}{cc} 2 & 1 \\ 2 & -1 \end{array}\right), B=\left(\begin{array}{lll} 2 & 1 & 2 \\ 2 & 2 & 1 \end{array}\right) . \]

1.5.1 Systems of algebraic equations

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Problem: Solve the system of equations \( \left\{\begin{array}{l}2 x+2 y=7 \\ 2 x-4 y=9\end{array}\right. \).

1.5.2 Systems of algebraic equations

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Problem: Solve the system of equations \[ \left\{\begin{array}{c} x_{1}-2 x_{2}-2 x_{3}+x_{4}=1 \\ 2 x_{1}+2 x_{2}-x_{3}+x_{4}=1 \\ x_{1}-x_{2}+x_{3}-2 x_{4}=2 \end{array} .\right. \]

1.5.3 Systems of algebraic equations

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Problem: Solve the matrix equation \( A X=B \), where \[ A=\left(\begin{array}{cccc} 0 & -5 & 0 & -2 \\ 2 & 1 & 1 & -4 \\ 0 & 0 & -5 & -1 \\ -1 & 1 & -3 & 2 \end{array}\right), B=\left(\begin{array}{c} -2 \\ -2 \\ -1 \\ 1 \end{array}\right) . \]

1.5.4 Systems of algebraic equations

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Problem: Solve the matrix equation using the Gauss method: \( A X=B \), where \[ A=\left(\begin{array}{ccccc} 2 & 0 & -3 & 15 & 14 \\ -2 & 3 & -2 & 4 & 1 \\ -2 & 2 & -4 & 16 & 18 \\ -4 & 2 & -2 & 5 & 10 \\ 3 & -1 & 3 & -13 & -19 \end{array}\right), B=\left(\begin{array}{c} 19 \\ 17 \\ 16 \\ 0 \\ -1 \end{array}\right) \text {. } \]

1.5.5 Systems of algebraic equations

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Problem: Solve the system of equations \( \left\{\begin{array}{c}3 x-y-z=2 \\ x+y+z=0 \\ 2 x+2 y+3 z=7\end{array}\right. \) a) Cramer's method, b) Gauss method, c) Inverse matrix method.

1.5.6 Systems of algebraic equations

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Problem: Solve the system of equations using the Cramer's, Gauss and inverse matrix methods: \[ \left\{\begin{array}{c} x+y-z=2 \\ 2 x+y=3 \\ x-2 y+z=0 \end{array} .\right. \]

1.5.7 Systems of algebraic equations

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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. \[ A=\left(\begin{array}{cccc} -2 & 3 & -1 & 0 \\ 2 & 0 & -2 & 0 \\ -6 & -3 & 3 & 0 \\ 0 & 0 & 0 & -2 \end{array}\right), B=\left(\begin{array}{cc} 1 & -2 \\ -3 & 0 \\ 2 & -1 \\ 1 & -1 \end{array}\right), A X=B . \]

1.5.8 Systems of algebraic equations

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

1.5.9 Systems of algebraic equations

5.09 $

Problem: Solve the homogeneous system of equations \[ \left\{\begin{array}{l} x_{1}+2 x_{2}+x_{3}+4 x_{4}+x_{5}=0 \\ 2 x_{1}+x_{2}+3 x_{3}+x_{4}-5 x_{5}=0 . \\ x_{1}+3 x_{2}-3 x_{3}+6 x_{4}-x_{5}=0 \end{array}\right. \]

1.5.10 Systems of algebraic equations

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Problem: Solve the system of equations \[ \left\{\begin{array}{c} x_{1}-x_{2}+4 x_{3}+3 x_{4}=0 \\ 3 x_{1}-2 x_{2}+x_{3}+2 x_{5}=1 \\ 2 x_{1}-x_{2}-3 x_{3}-3 x_{4}+2 x_{5}=1 \end{array}\right. \]

1.5.11 Systems of algebraic equations

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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. \[ \begin{array}{l} A=\left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 83 & -47 & 1 & 0 & 0 \\ -55 & 94 & 0 & 1 & 0 \\ 62 & -71 & 0 & 0 & 1 \end{array}\right), \\ B=\left(\begin{array}{ccccc} 2 & -3 & 4 & -1 & 0 \\ 4 & -2 & 1 & 0 & 3 \end{array}\right), \quad X A=B . \end{array} \]

1.5.12 Systems of algebraic equations

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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. \[ \left\{\begin{array}{c} -3 x_{1}+5 x_{2}-6 x_{3}=-5 \\ 2 x_{1}-3 x_{2}+5 x_{3}=8 \\ x_{1}+4 x_{2}-x_{3}=1 \end{array}\right. \]

1.5.13 Systems of algebraic equations

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Problem: Given a 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. \[ \left\{\begin{array}{c} 2 x_{1}+4 x_{2}-3 x_{3}=-10 \\ -x_{1}+5 x_{2}-2 x_{3}=5 \\ 3 x_{1}-2 x_{2}+4 x_{3}=3 \end{array}\right. \]

1.5.14 Systems of algebraic equations

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