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Problem: Find the decomposition of the vector \( a=(-2 ; 2 ; 2) \) into vectors \( b=(2 ; 1 ; 2) \), \( c=(0 ; 2 ;-2), d=(1 ; 2 ; 0) \).

1.1.1 Vector Algebra

0.72 $

Problem: 1. Find the dot product of vectors \( a=(\Gamma ; H ; 1 ; \Gamma ; 2) \) and \( b=(3 ; \Gamma ; 2 ; H ; 1) \). 2. Find the angle between vectors \( a=(1 ; H ; \Gamma ; 1 ; 2) \) and \( b=(2 ; H ; \Gamma ; 3 ; 1) \). 3. Find the cross product of vectors \( a=(1 ; H ; \Gamma) \) and \( b=(2 ; H ; \Gamma) \). Where \( \Gamma=2, H=2 \).

1.1.2 Vector Algebra

2.41 $

Problem: Find inverse matrix to matrix \( A \) of size \( \times n \) \[ \left(\begin{array}{cccccc} n & n-1 & n-2 & \ldots & 2 & 1 \\ 1 & n-1 & n-2 & \ldots & 2 & 1 \\ 1 & 1 & n-2 & \ldots & 2 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 1 & 1 & \ldots & 2 & 1 \\ 1 & 1 & 1 & \ldots & 1 & 1 \end{array}\right) . \]

1.4.1 Matrix transformations

3.62 $

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

0.72 $

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

0.48 $

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

0.96 $

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

2.41 $

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

2.41 $

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

2.41 $

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

2.41 $

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

4.82 $

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

4.82 $

Problem: 1. Find the angle \( C \) in the triangle \( A B C \), where \( A=(2 ; \Gamma ; H), B=(3 ; H ; \Gamma), C=(-3 ; 1 ; \Gamma) \). 2. Find the area of the triangle \( A B C \). 3. Find the volume of the tetrahedron \( A B C D \), where \( \quad A=(1 ; \Gamma ; H), \quad B=(-3 ; H ; \Gamma) \), \( C=(\Gamma ; 3 ;-1), D=(2 ; 7 ; \Gamma) \). Where \( \Gamma=2, H=2 \).

1.1.3 Vector Algebra

2.41 $

Problem: Calculate the determinant \( \left|\begin{array}{lllll}0 & 1 & 2 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 & 0\end{array}\right| \).

1.2.1 Determinant calculation

0.72 $

Problem: Calculate the determinant of the matrix of the \( n \)-th order \( a_{j k}=|j-k| \).

1.2.2 Determinant calculation

4.82 $

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