Collection of Maths Problems

Matrices of order 5

Task number: 2598

• Variant 1

  \(\begin{pmatrix} 3 & 1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 & 1 \\ 1 & 1 & 3 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 & 3 \\ \end{pmatrix} \)

• Result

  The determinant of the matrix is \(36\).

• Variant 2

  \(\begin{pmatrix} 1 & 2 & 0 & 2 & -1 \\ 0 & 1 & 3 & -1 & 0 \\ 1 & 2 & 0 & 2 & 1 \\ 0 & -1 & 3 & 1 & 0 \\ -1 & 2 & 0 & 2 & 1 \\ \end{pmatrix} \)

• Result

  The determinant of the matrix is \(96\).

• Variant 3

  \(\begin{pmatrix} 1 & -1 & 1 & 1 & 3 \\ 0 & 2 & 1 & 0 & -1 \\ 2 & -1 & 1 & -3 & 0 \\ 0 & 0 & -1 & 2 & 1 \\ 3 & 2 & 0 & 1 & 0 \\ \end{pmatrix} \)

• Result

  The determinant of the matrix is \(10\).

• Variant 4

  \(\begin{pmatrix} 0 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 2 \\ 1 & 2 & 3 & 2 & 1 \\ 2 & 3 & 2 & 1 & 0 \\ 3 & 2 & 1 & 0 & 0 \\ \end{pmatrix} \)

• Result

  The determinant of the matrix is \(16\).

• Variant 5

  \(\begin{pmatrix} 1 & 1 & -1 & 1 & 1 \\ -1 & 1 & 1 & 1 & -1 \\ 1 & -1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 & -1 \\ -1 & 1 & 1 & -1 & 1 \\ \end{pmatrix} \)

• Result

  The determinant of the matrix is \(-16\).

• Variant 6

  \(\begin{pmatrix} 2 & 0 & 2 & 0 & -2 \\ 0 & 3 & 0 & 2 & 0 \\ 2 & 0 & -2 & 0 & 2 \\ 0 & 2 & 0 & 3 & 0 \\ -2 & 0 & 2 & 0 & 2 \\ \end{pmatrix} \)

• Hint

  Rearrange rows and columns.

• Resolution

  \(\begin{vmatrix} 2 & 0 & 2 & 0 & -2 \\ 0 & 3 & 0 & 2 & 0 \\ 2 & 0 & -2 & 0 & 2 \\ 0 & 2 & 0 & 3 & 0 \\ -2 & 0 & 2 & 0 & 2 \\ \end{vmatrix} = \begin{vmatrix} 2 & 2 &-2 & 0 & 0 \\ 2 &-2 & 2 & 0 & 0 \\ -2 & 2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 2\\ 0 & 0 & 0 & 2 & 3\\ \end{vmatrix} = \begin{vmatrix} 2 & 2 &-2\\ 2 &-2 & 2\\ -2 & 2 & 2\\ \end{vmatrix}\cdot \begin{vmatrix} 3 & 2\\ 2 & 3\\ \end{vmatrix} \)

• Result

  The determinant of the matrix is \(-160\).