Solve for the determinants: $\begin{vmatrix}1 & 2 & 3 \\ 2 & -1 & 4 \\-1 & -1 & -5\end{vmatrix}$

$D = \begin{vmatrix} 1 & 2 & 3 \\ 2 & -1 & 4 \\ -1 & -1 & -5 \end{vmatrix}$
 

$D = \left| \begin{matrix} 1 & 2 & 3 \\ 2 & -1 & 4 \\ -1 & -1 & -5 \\ \end{matrix} \left| \, \begin{matrix} 1 & 2 \\ 2 & -1 \\ -1 & -1 \\ \end{matrix} \right. \right|$

$D = \text{Sum of product of downward diagonals} - \text{Sum of product of upward diagonals}$

$D = \left[ 1(-1)(-5) + 2(4)(-1) + 3(2)(-1) \right] - \left[ -1(-1)(3) + (-1)(4)(1) + (-5)(2)(2) \right]$

$D = (5 - 8 - 6) - (3 - 4 -20)$

$D = -9 - (-21)$

$D = -9 + 21$

$D = 12$
 

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Use the matrix function of your calculator