Problems from General method for the calculation of determinants

Invent a $4 \times 4$ matrix and compute its determinant.

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Development:

$C = (\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 3 & 1 & - 1 \\ 1 & 0 & 1 & 2 \\ 0 & 1 & 1 & 0 \end{matrix})$

$d e t (C) = 1 \cdot d e t (B) + 1 \cdot | \begin{matrix} 0 & 3 & - 1 \\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{matrix} | =$

$1 \cdot (- 3) + 1 \cdot [0 \cdot 0 \cdot 0 + 1 \cdot 1 \cdot (- 1) + 0 \cdot 2 \cdot 3 - 0 \cdot 0 \cdot (- 1) - 1 \cdot 2 \cdot 0 - 0 \cdot 1 \cdot 3] =$

$= - 3 + (0 - 1 + 0 - 0 - 0 - 0) = - 3 - 1 = - 4$

Solution:

$d e t (C) = - 2$ .

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