Evaluate the determinant.
-$$\left| \begin{array} { l l l l }
0 & 7 & 7 & 3 \
0 & 2 & 5 & 6 \
5 & 7 & 9 & 4 \
6 & 2 & 2 & 4
\end{array} \right|$$
A) 636
B) 27
C) 1
D) 22

Question

Evaluate the determinant.
-$$\left| \begin{array} { l l l l } 
0 & 7 & 7 & 3 \
0 & 2 & 5 & 6 \
5 & 7 & 9 & 4 \
6 & 2 & 2 & 4
\end{array} \right|$$
A) 636
B) 27
C) 1
D) 22

Quizplus · Accepted Answer

The determinant of a matrix can be calculated using various methods, such as expansion by minors or row reduction. Given the structure of this matrix, where the first column has zeros except for the third and fourth rows, it's efficient to expand along the first column. This leads to:$$ \left| \begin{array}{lll} 0 & 7 & 7 & 3 \ 0 & 2 & 5 & 6 \ 5 & 7 & 9 & 4 \ 6 & 2 & 2 & 4 \end{array} \right| = 5 \cdot (-1)^{3+1} \cdot \left| \begin{array}{lll} 7 & 7 & 3 \ 2 & 5 & 6 \ 2 & 2 & 4 \end{array} \right| + 6 \cdot (-1)^{4+1} \cdot \left| \begin{array}{lll} 7 & 7 & 3 \ 2 & 5 & 6 \ 7 & 9 & 4 \end{array} \right| $$Calculating each of these 3x3 determinants (which can be done using the rule of Sarrus or further expansion) and summing them up with the appropriate signs gives the total determinant. The correct calculation results in a determinant of 636.

Evaluate the Determinant ∣0773025657946224∣\left| \begin{array} { l l l l } 0 & 7 & 7 & 3 \\0 & 2 & 5 & 6 \\5 & 7 & 9 & 4 \\6 & 2 & 2 & 4\end{array} \right|​0056​7272​7592​3644​​

Evaluate the Determinant $\left| \begin{array} { l l l l } 0 & 7 & 7 & 3 \\0 & 2 & 5 & 6 \\5 & 7 & 9 & 4 \\6 & 2 & 2 & 4\end{array} \right|$