Collection of Maths Problems

Determinant expansion

Task number: 4437

• Variant

  Over \(\mathbb Z_5\): \(\begin{pmatrix} 1 & 2 & 3 \\ 4 & 4 & 1 \\ 2 & 3 & 3 \\ \end{pmatrix} \)

• Solution

  \(\begin{vmatrix} 1 & 2 & 3 \\ 4 & 4 & 1 \\ 2 & 3 & 3 \\ \end{vmatrix} = -4 \begin{vmatrix} 2 & 3 \\ 3 & 3 \\ \end{vmatrix} + 4 \begin{vmatrix} 1 & 3 \\ 2 & 3 \\ \end{vmatrix} - \begin{vmatrix} 1 & 2 \\ 2 & 3 \\ \end{vmatrix} = 2+3+1=1\)

• Varianta

  Over \(\mathbb R\): \(\begin{pmatrix} 0 & 1 & 2 & 0 \\ 3 & 0 & 0 & 4 \\ 0 & 5 & 6 & 0 \\ 7 & 0 & 0 & 8 \\ \end{pmatrix} \)

• Solution

  \(\begin{vmatrix} 0 & 1 & 2 & 0 \\ 3 & 0 & 0 & 4 \\ 0 & 5 & 6 & 0 \\ 7 & 0 & 0 & 8 \\ \end{vmatrix} = -3 \begin{vmatrix} 1 & 2 & 0 \\ 5 & 6 & 0 \\ 0 & 0 & 8 \\ \end{vmatrix} +0 \begin{vmatrix} 0 & 2 & 0 \\ 0 & 6 & 0 \\ 7 & 0 & 8 \\ \end{vmatrix} -0 \begin{vmatrix} 0 & 1 & 0 \\ 0 & 5 & 0 \\ 7 & 0 & 8 \\ \end{vmatrix} +4 \begin{vmatrix} 0 & 1 & 2 \\ 0 & 5 & 6 \\ 7 & 0 & 0 \\ \end{vmatrix} = 96-112=-16\)

• Varianta

  Over \(\mathbb R\): \(\begin{pmatrix} 0 & 1 & 2 & 0 \\ 3 & 100 & 100 & 4 \\ 0 & 5 & 6 & 0 \\ 7 & 100 & 100 & 8 \\ \end{pmatrix}\)

• Solution

  \(\begin{vmatrix} 0 & 1 & 2 & 0 \\ 3 & 100 & 100 & 4 \\ 0 & 5 & 6 & 0 \\ 7 & 100 & 100 & 8 \\ \end{vmatrix} = -3 \begin{vmatrix} 1 & 2 & 0 \\ 5 & 6 & 0 \\ 100 & 100 & 8 \\ \end{vmatrix} +100 \begin{vmatrix} 0 & 2 & 0 \\ 0 & 6 & 0 \\ 7 & 0 & 8 \\ \end{vmatrix} -100 \begin{vmatrix} 0 & 1 & 0 \\ 0 & 5 & 0 \\ 7 & 0 & 8 \\ \end{vmatrix} +4 \begin{vmatrix} 0 & 1 & 2 \\ 0 & 5 & 6 \\ 7 & 100 & 100 \\ \end{vmatrix} =96-112=-16\)