Which of the following evaluates the determinant $\left| \begin{array} { c c c }
6 & - 3 & 4 \
5 & 7 & - 5 \
8 & 2 & 6
\end{array} \right|$ by expanding by row 2?
A) $- 5 \left| \begin{array} { c c }
- 3 & 4 \
2 & 6
\end{array} \right| + 7 \left| \begin{array} { c c }
6 & 4 \
8 & 6
\end{array} \right| + 5 \left| \begin{array} { c c }
6 & - 3 \
8 & 2
\end{array} \right|$
B) $\left| \begin{array} { c c }
- 3 & 4 \
2 & 6
\end{array} \right| - 7 \left| \begin{array} { c c }
6 & 4 \
8 & 6
\end{array} \right| - 5 \left| \begin{array} { c c }
6 & - 3 \
8 & 2
\end{array} \right|$
C) $- 5 \left| \begin{array} { c c }
- 3 & 4 \
2 & 6
\end{array} \right| + 7 \left| \begin{array} { c c }
6 & 4 \
8 & 6
\end{array} \right| - 5 \left| \begin{array} { c c }
6 & - 3 \
8 & 2
\end{array} \right|$
D) none of these

Question

Which of the following evaluates the determinant $\left| \begin{array} { c c c } 
6 & - 3 & 4 \
5 & 7 & - 5 \
8 & 2 & 6
\end{array} \right|$ by expanding by row 2?
A) $- 5 \left| \begin{array} { c c } 
- 3 & 4 \
2 & 6
\end{array} \right| + 7 \left| \begin{array} { c c } 
6 & 4 \
8 & 6
\end{array} \right| + 5 \left| \begin{array} { c c } 
6 & - 3 \
8 & 2
\end{array} \right|$
B) $\left| \begin{array} { c c } 
- 3 & 4 \
2 & 6
\end{array} \right| - 7 \left| \begin{array} { c c } 
6 & 4 \
8 & 6
\end{array} \right| - 5 \left| \begin{array} { c c } 
6 & - 3 \
8 & 2
\end{array} \right|$
C) $- 5 \left| \begin{array} { c c } 
- 3 & 4 \
2 & 6
\end{array} \right| + 7 \left| \begin{array} { c c } 
6 & 4 \
8 & 6
\end{array} \right| - 5 \left| \begin{array} { c c } 
6 & - 3 \
8 & 2
\end{array} \right|$
D) none of these

Quizplus · Accepted Answer

Expanding by row 2 involves multiplying each element of row 2 by the determinant of the matrix that remains after removing the row and column of that element, with alternating signs starting with a positive sign for the top-left element. The correct expansion is: $-5$ times the determinant of the 2x2 matrix obtained by removing row 2 and column 1, $+7$ times the determinant of the 2x2 matrix obtained by removing row 2 and column 2, and $+5$ times the determinant of the 2x2 matrix obtained by removing row 2 and column 3, which matches option A.

Which of the Following Evaluates the Determinant ∣6−3457−5826∣\left| \begin{array} { c c c } 6 & - 3 & 4 \\5 & 7 & - 5 \\8 & 2 & 6\end{array} \right|​658​−372​4−56​​

Which of the Following Evaluates the Determinant $\left| \begin{array} { c c c } 6 & - 3 & 4 \\5 & 7 & - 5 \\8 & 2 & 6\end{array} \right|$