Question 10.4.2: Evaluating a 3 × 3 Determinant Evaluate the determinant |1 2...

Evaluating \text { a } 3 \times 3 Determinant

Evaluate the determinant \left|\begin{array}{rrr}1 & 2 & 4 \\-3 & 3 & 1 \\3 & -2 & 5\end{array}\right|.

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Expanding along the first row, we have:

\left|\begin{array}{rrr}1 & 2 & 4 \\-3 & 3 & 1 \\3 & -2 & 5\end{array}\right|=(1)\left|\begin{array}{rr}3 & 1 \\-2 & 5\end{array}\right|-(2)\left|\begin{array}{rr}-3 & 1 \\3 & 5\end{array}\right|+(4)\left|\begin{array}{rr}-3 & 3 \\3 & -2\end{array}\right|

= (1)[(3)(5) − (1)(−2)] − (2)[(−3)(5) − (1)(3)]

+ (4)[(−3)(−2) − (3)(3)]

= 41.

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