Evaluate B left begin matrix 2 amp 3 amp - 4 1 ... - SS3 Mathematics Matrices and Determinants Question

Evaluate $| B | = | \begin{matrix} 2 & 3 & - 4 \\ 1 & 2 & 3 \\ 3 & - 1 & - 1 \end{matrix} |$

Using the third row, $| A | = a_{31} C_{31} + a_{32} C_{32} + a_{33} C_{33}$

$C_{31} = {(- 1)}^{3 + 1} | \begin{matrix} 3 & - 4 \\ 2 & 3 \end{matrix} | = (- 1)^{4} | \begin{matrix} 3 & - 4 \\ 2 & 3 \end{matrix} | = + [3 (3) - (- 4) 2] = + [9 + 8] = 17$

$C_{12} = {(- 1)}^{3 + 2} | \begin{matrix} 2 & - 4 \\ 1 & 3 \end{matrix} | = (- 1)^{5} | \begin{matrix} 2 & - 4 \\ 1 & 3 \end{matrix} | = - [2 (3) - (- 4) (1)] = - [6 + 4] = - 10$

$C_{13} = {(- 1)}^{3 + 3} | \begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix} | = (- 1)^{6} | \begin{matrix} 2 & 3 \\ 1 & 2 \end{matrix} | = + [2 (2) - 3 (1)] = + [4 - 3] = 1$

$| A | = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13} = 3 (17) + (- 1) (- 10) + (- 1) 1 = 51 + 10 - 1 = 60$

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