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Problem: Calculate for all \( n \) the determinant of the matrix \( A=\left(a_{j k}\right) \quad \) of the order \( n \), where \( a_{j k}=2 j k^{2}-k j^{2}-1 \).

1.2.3 Determinant calculation

5.1 $

Problem: Calculate the determinant of a matrix of size \[ \times n\left(\begin{array}{cccccc} 1 & 0 & \ldots & \ldots & 0 & 2 \\ 0 & 1 & \ldots & \ldots & 2 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 2 & \ldots & \ldots & 1 & 0 \\ 2 & 0 & \ldots & \ldots & 0 & 1 \end{array}\right) . \]

1.2.4 Determinant calculation

3.82 $

Problem: Calculate the determinant of a matrix of size \[ \times n\left(\begin{array}{ccccc} 2 & 1 & \ldots & \ldots & 1 \\ 1 & 2 & \ldots & \ldots & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 1 & \ldots & \ldots & 2 \end{array}\right) \text {. } \]

1.2.5 Determinant calculation

2.55 $

Problem: Calculate the determinant of the \( n \)-th order \[ \Delta=\left|\begin{array}{cccccc} 0 & a_{2} & a_{3} & \ldots & a_{n-1} & a_{n} \\ b_{1} & 0 & a_{3} & \ldots & a_{n-1} & a_{n} \\ b_{1} & b_{2} & 0 & \ldots & a_{n-1} & a_{n} \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ b_{1} & b_{2} & b_{3} & \ldots & 0 & a_{n} \\ b_{1} & b_{2} & \ldots & \ldots & b_{n-1} & 0 \end{array}\right| . \]

1.2.6 Determinant calculation

5.1 $

Problem: Calculate the determinant of the \( n \)-th order \[ \Delta=\left|\begin{array}{ccccccc} 1 & 2 & 3 & \ldots & \ldots & n-1 & n \\ 1 & 3 & 3 & 4 & \ldots & n-1 & n \\ 1 & 2 & 5 & 4 & \ldots & n-1 & n \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 2 & 3 & \ldots & \ldots & 2 n-3 & n \\ 1 & 2 & 3 & \ldots & \ldots & n-1 & 2 n-1 \end{array}\right| . \]

1.2.7 Determinant calculation

2.55 $

Problem: Calculate the determinant of \( (n+1) \)-th order \[ \Delta=\left|\begin{array}{ccccc} x & y_{1} & y_{2} & \ldots & y_{n} \\ y_{1} & x & y_{2} & \ldots & y_{n} \\ y_{1} & y_{2} & x & \ldots & y_{n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ y_{1} & y_{2} & y_{3} & \ldots & x \end{array}\right| \text {. } \]

1.2.8 Determinant calculation

3.82 $

Problem: Calcualte the determinant of \( n \)-th order \[ \Delta_{n}=\left|\begin{array}{cccccc} a+b & a b & 0 & \ldots & 0 & 0 \\ 2 & a+b & a b & \ldots & 0 & 0 \\ 0 & 1 & a+b & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & \ldots & a+b & a b \\ 0 & 0 & \ldots & \ldots & 1 & a+b \end{array}\right| . \]

1.2.9 Determinant calculation

5.1 $

Problem: Solve the equation in the permutation group \( S_{\mathcal{T}}:(1234567) \cdot X \cdot(26)=(134) \cdot(275) \). Expand the permutation into cycles and transpositions.

1.3.1 Permutation group

2.04 $

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

1.02 $

Problem: Construct non-isomorphic groups with 3-element base set.

1.6.1 Fields, Groups, Rings

2.04 $

Problem: Find the matrix centralizer \( A=\left(\begin{array}{ll}3 & -4 \\ 1 & -2\end{array}\right) \) in \( G L(2, R) \), where \( G L(n, F) \) is a group of non-degenerate matrices of size \( n \times n \) over the field \( F \), with the operation of matrix multiplication.

1.6.2 Fields, Groups, Rings

3.06 $

Problem: Find out if the transformations are linear: \[ \begin{array}{l} A x=\left(3 x_{1}-2 x_{2}-x_{3}, 1, x_{1}+2 x_{2}+3\right), \\ B x=\left(3 x_{1}-2 x_{2}-x_{3}, 1, x_{1}^{3}+2 x_{2}+3 x_{3}\right), \\ C x=\left(3 x_{1}-2 x_{2}-x_{3}, x_{3}, x_{1}+2 x_{2}+3 x_{3}\right) . \end{array} \]

1.7.1 Linear transformations

1.27 $

Problem: Investigate linear dependence of the vectors \( e^{x}, e^{2 x}, e^{3 x} \) in \( (-\infty,+\infty) \).

8.1.3.28 Higher order differential equations

1.02 $

Problem: Given the coordinates of the vector \( x=\{2,5,10\} \) in the basis \( \left(e_{1}, e_{2}, e_{3}\right) \). Find its coordinates in the basis \( \left(e_{1}{ }^{\prime}, e_{2}{ }^{\prime}, e_{3}{ }^{\prime}\right) \), where \[ \left\{\begin{array}{c} e_{1}{ }^{\prime}=e_{1}+e_{2}+6 e_{3} \\ e_{2}{ }^{\prime}=\frac{6}{5} e_{1}-e_{2} \\ e_{3}{ }^{\prime}=-e_{1}+e_{2}+e_{3} \end{array} .\right. \]

1.7.3 Linear transformations

1.27 $

Problem: Complement the system of vectors \( e_{1}=\{1,2,3,4\}, \quad e_{2}=\{-1,-3,1,1\} \quad \) to the orthogonal system in \( R^{4} \).

1.7.4 Linear transformations

1.27 $

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