Question
Evaluate the following determinant:
$$ det \left( \begin{matrix}8&3&4\\-4&5&1\\3&7&9\end{matrix} \right) $$Answer
$$ det \left( \begin{matrix}8&3&4\\-4&5&1\\3&7&9\end{matrix} \right) = 249 $$
Explanation
To find the 3x3 determinant, we can use the Rule of Sarrus. .
$$ \begin{aligned} det \left( \begin{matrix}8&3&4\\-4&5&1\\3&7&9\end{matrix} \right) &= \left[
\begin{array}{ccc|cc} \cssId{i00}{8} & \cssId{i01}{3} & \cssId{i02}{4} & \cssId{i03}{8} & \cssId{i04}{3} \\
\cssId{i10}{-4} & \cssId{i11}{5} & \cssId{i12}{1} & \cssId{i13}{-4} & \cssId{i14}{5} \\
\cssId{i20}{3} & \cssId{i21}{7} & \cssId{i22}{9} & \cssId{i23}{3} & \cssId{i24}{7}
\end{array} \right | = \\\\
&= \cssId{u0}{8 \cdot 5 \cdot 9} \cssId{u1}{+3 \cdot 1 \cdot 3} \cssId{u2}{+4 \cdot \left(-4\right) \cdot 7} \cssId{u3}{-3 \cdot 5 \cdot 4} \cssId{u4}{-7 \cdot 1 \cdot 8} \cssId{u5}{-9 \cdot \left(-4\right) \cdot 3} = \\\\
&= 360 + 9 + \left( -112\right) - 60 - 56 - \left( -108\right) = 249
\end{aligned}
$$
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