Question
Evaluate:
$\left|\begin{array}{ccc}1 & -3 & 12 \\ 0 & 2 & -4 \\ 9 & 7 & 2\end{array}\right|$
$\left|\begin{array}{ccc}1 & -3 & 12 \\ 0 & 2 & -4 \\ 9 & 7 & 2\end{array}\right|$
$=1\left|\begin{array}{rr}2 & -4 \\ 7 & 2\end{array}\right|-(-3)\left|\begin{array}{rr}0 & -4 \\ 9 & 2\end{array}\right|+12\left|\begin{array}{ll}0 & 2 \\ 9 & 7\end{array}\right|$
= 1(4 + 28) + 3(0 + 36) + 12(0 – 18) = 1(32) + 3(36) + 12(-18) = 32 + 108 – 216 = -76
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$\begin{aligned} & {\left[\begin{array}{lll}0 & -1 & 4\end{array}\right]\left\{2\left[\begin{array}{cc}4 & 5 \\ 3 & 6 \\ 2 & -1\end{array}\right]+3\left[\begin{array}{cc}4 & 3 \\ 1 & 4 \\ 0 & -1\end{array}\right]\right\}} \\ & =\left[\begin{array}{ll}x & y\end{array}\right] .\end{aligned}$
$\frac{x}{3}-\frac{y}{2}=0$