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Problem: Calculate the determinant \( \Delta_{n} \) of size \( n \times n \). \[ \Delta_{n}=\left|\begin{array}{ccccc} x & y & 0 & \cdots & 0 \\ 0 & x & y & \cdots & 0 \\ 0 & 0 & x & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ y & 0 & 0 & \cdots & x \end{array}\right| \text {. } \]

1.2.10 Determinant calculation

2.4 $

Problem: Calculate the determinant \( \Delta_{n} \) of size \( n \times n \). \[ \Delta_{n}=\left|\begin{array}{ccccc} x+1 & x & x & \cdots & x \\ x & x+2 & x & \cdots & x \\ x & x & x+3 & \cdots & x \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ x & x & x & \cdots & x+n \end{array}\right| . \]

1.2.11 Determinant calculation

4.8 $

Problem: Calculate the determinant of the order \( (n+1) \times(n+1) \). \[ \Delta_{n+1}=\left|\begin{array}{cccccc} 0 & 1 & 1 & \cdots & 1 & 1 \\ 1 & 0 & a_{1}+a_{2} & \cdots & a_{1}+a_{n-1} & a_{1}+a_{n} \\ 1 & a_{2}+a_{1} & 0 & \cdots & a_{2}+a_{n-1} & a_{2}+a_{n} \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 1 & a_{n-1}+a_{1} & a_{n-1}+a_{2} & \cdots & 0 & a_{n-1}+a_{n} \\ 1 & a_{n}+a_{1} & a_{n}+a_{2} & \cdots & a_{n}+a_{n-1} & 0 \end{array}\right| . \]

1.2.12 Determinant calculation

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Problem: Calculate the determinant \( \Delta_{n} \) of order \( n \times n \). \[ \Delta_{n}=\left|\begin{array}{ccccc} 1 & 1 & \cdots & 1 & -n \\ 1 & 1 & \cdots & -n & 1 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 1 & -n & \cdots & 1 & 1 \\ -n & 1 & \cdots & 1 & 1 \end{array}\right| \]

1.2.13 Determinant calculation

2.4 $

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

4.8 $

Problem: For a given group \( G \) and its subgroup \( H \) : a) describe (left) cosets, b) find out if the factor group is defined, c) if the factor group is defined, give an example of an isomorphic group. \[ G=(G L(n, \mathbb{R}) \cdot \cdot), H=\{A \in G L(n \cdot \mathbb{R}):|\operatorname{det} A|=1\} . \]

1.6.8 Fields, Groups, Rings

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Problem: Find out whether the set of integers divisible by 5 , form a group with respect to the addition. \[ M=\{n \in \mathbb{Z} \mid n: 5\}=\{5 k \mid k \in \mathbb{Z}\} . \]

1.6.23 Fields, Groups, Rings

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Problem: Find the ideal (10) in rings \[ (\mathbb{Z} ;+; \cdot) ;(\mathbb{Z}[i] ;+; \cdot) ;(\mathbb{Z}[x] ;+; \cdot) ;(\mathbb{Q}[x] ;+; \cdot) \text {. } \]

1.6.24 Fields, Groups, Rings

1.92 $

Problem: Prove that the groups \( (\mathbb{Z} ;+) \) and \( (G ; \cdot) \) are isomorphic; where \( G=\left\{x \mid x=2^{t}, t \in \mathbb{Z}\right\} \).

1.6.25 Fields, Groups, Rings

1.92 $

Problem: Prove that numbers \( 5 ;-5 ; 5 i ;-5 i \) are reducible in the \( \operatorname{ring}(\mathbb{Z}[i] ;+; \cdot) \).

1.6.26 Fields, Groups, Rings

2.4 $

Problem: An algebraic system \( (X, *) \) with one binary operation "*" is given. Indicate what type (groupoid, semigroup, monoid, group) this system belongs to.

1.6.27 Fields, Groups, Rings

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Problem: For the given ring \( K \) and its subring \( H \) : a) describe (left) cosets; b) find out whether the quotient ring is defined; c) if a quotient ring is defined, give an example of an isomorphic ring; \( K=\left(\mathbb{R}^{\infty},+, \cdot\right) \) is a ring of all sequences of real numbers, \( \quad H=\left\{\left\{x_{n}\right\} \in \mathbb{R}^{\infty}: x_{1}=\cdots=x_{n}=0\right\} \) ( \( n \) is fixed).

1.6.28 Fields, Groups, Rings

2.88 $

Problem: Find out whether the set of powers of number 7 with integer exponents form a group with respect to multiplication. \[ M=\left\{7^{n} \mid n \in \mathbb{Z}\right\} \text {. } \]

1.6.29 Fields, Groups, Rings

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Problem: Let \( H \) be a subgroup, generated by the element \( b \) in the multiplicative \( \mathbb{Z}_{p}^{*} \) of residues modulo \( p \), and the \( g H \) coset of the subgroup \( H \), generates by the element \( g \). Calculate the subgroup \( H \) and the coset \( g H \). \[ p=97, b=8, g=2 \text {. } \]

1.6.30 Fields, Groups, Rings

2.88 $

Problem: Solve the system of equations in finite fields \( \mathbb{Z}_{7} \) and \( \mathbb{Z}_{11} \) : \[ \left\{\begin{array}{l} 5 x+3 y=1 \\ 3 x-2 y=2 \end{array}\right. \]

1.10.4 Polynomials

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