Evaluate $3 \times 3$ determinant. $$\left| \begin{array} { c c c }
1 & 4 & - 1 \
5 & 1 & 4 \
4 & 6 & 5
\end{array} \right|$$ Use the properties of determinants to your advantage.
A) -74
B) -7
C) -81

Question

Evaluate $3 \times 3$ determinant. $$\left| \begin{array} { c c c } 
1 & 4 & - 1 \
5 & 1 & 4 \
4 & 6 & 5
\end{array} \right|$$ Use the properties of determinants to your advantage.
A) -74
B) -7
C) -81

Quizplus · Accepted Answer

To evaluate the determinant, we can apply the expansion by minors method along the first row. This gives us $$\left| \begin{array} { c c c } 
1 & 4 & - 1 \
5 & 1 & 4 \
4 & 6 & 5
\end{array} \right| = 1\left| \begin{array} { c c } 
1 & 4 \
6 & 5
\end{array} \right| -4\left| \begin{array} { c c } 
5 & 4 \
6 & 5
\end{array} \right| -1\left| \begin{array} { c c } 
5 & 1 \
6 & 4
\end{array} \right|.$$ Evaluating these three $2\times 2$ determinants gives us $$1\cdot(-19)-4\cdot(-14)-(-5)=\boxed{-81}$$ as our final answer.

Evaluate 3×33 \times 33×3 Determinant ∣14−1514465∣\left| \begin{array} { c c c } 1 & 4 & - 1 \\5 & 1 & 4 \\4 & 6 & 5\end{array} \right|​154​416​−145​​

Evaluate $3 \times 3$ Determinant $\left| \begin{array} { c c c } 1 & 4 & - 1 \\5 & 1 & 4 \\4 & 6 & 5\end{array} \right|$