Determinant calculation/Determinant calculation

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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

0.76 $

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

1.2.2 Determinant calculation

5.08 $

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

5.08 $

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

3.81 $

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

2.54 $

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

5.08 $

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

2.54 $

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

3.81 $

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

5.08 $

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

2.54 $

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

5.08 $

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

6.35 $

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

2.54 $