Evaluate the determinant.
-$$\left| \begin{array} { r r r r }
3 & - 2 & 2 & - 1 \
4 & 1 & 1 & 6 \
- 3 & 2 & - 2 & 1 \
5 & 0 & 3 & - 2
\end{array} \right|$$
A) 6
B) 37
C) 9
D) 0

Question

Evaluate the determinant.
-$$\left| \begin{array} { r r r r } 
3 & - 2 & 2 & - 1 \
4 & 1 & 1 & 6 \
- 3 & 2 & - 2 & 1 \
5 & 0 & 3 & - 2
\end{array} \right|$$
A) 6
B) 37
C) 9
D) 0

Quizplus · Accepted Answer

The determinant of a matrix can be calculated using various methods, including expansion by minors or row reduction. For this matrix, one way to see that the determinant is 0 without fully calculating it is to notice that the first and third rows are proportional. Specifically, the third row is the first row multiplied by -1 ($3*(-1) = -3$, $-2*(-1) = 2$, $2*(-1) = -2$, $-1*(-1) = 1$). When a matrix has two proportional rows, its determinant is 0.

Evaluate the Determinant ∣3−22−14116−32−21503−2∣\left| \begin{array} { r r r r } 3 & - 2 & 2 & - 1 \\4 & 1 & 1 & 6 \\- 3 & 2 & - 2 & 1 \\5 & 0 & 3 & - 2\end{array} \right|​34−35​−2120​21−23​−161−2​​

Evaluate the Determinant $\left| \begin{array} { r r r r } 3 & - 2 & 2 & - 1 \\4 & 1 & 1 & 6 \\- 3 & 2 & - 2 & 1 \\5 & 0 & 3 & - 2\end{array} \right|$