Question
Evaluate the following determinant:
$$ det \left( \begin{matrix}13&5&8\\2&-1&23\\5&29&-4\end{matrix} \right) $$Answer
$$ det \left( \begin{matrix}13&5&8\\2&-1&23\\5&29&-4\end{matrix} \right) = -7500 $$
Explanation
To find the 3x3 determinant, we can use the Rule of Sarrus. .
$$ \begin{aligned} det \left( \begin{matrix}13&5&8\\2&-1&23\\5&29&-4\end{matrix} \right) &= \left[
\begin{array}{ccc|cc} \cssId{i00}{13} & \cssId{i01}{5} & \cssId{i02}{8} & \cssId{i03}{13} & \cssId{i04}{5} \\
\cssId{i10}{2} & \cssId{i11}{-1} & \cssId{i12}{23} & \cssId{i13}{2} & \cssId{i14}{-1} \\
\cssId{i20}{5} & \cssId{i21}{29} & \cssId{i22}{-4} & \cssId{i23}{5} & \cssId{i24}{29}
\end{array} \right | = \\\\
&= \cssId{u0}{13 \cdot \left(-1\right) \cdot \left(-4\right)} \cssId{u1}{+5 \cdot 23 \cdot 5} \cssId{u2}{+8 \cdot 2 \cdot 29} \cssId{u3}{-5 \cdot \left(-1\right) \cdot 8} \cssId{u4}{-29 \cdot 23 \cdot 13} \cssId{u5}{-\left(-4\right) \cdot 2 \cdot 5} = \\\\
&= 52 + 575 + 464 - \left( -40\right) - 8671 - \left( -40\right) = -7500
\end{aligned}
$$
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