Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } - 3 & 2 & - 8 \ 3 & - 2 & 6 \ - 1 & 3 & - 6 \end{array} \right]$.
A) $M _ { 13 } = \left| \begin{array} { r r } - 3 & - 8 \ - 1 & - 6 \end{array} \right| = 10$
$$C _ { 13 } = - 10$$
B) $M _ { 13 } = \left| \begin{array} { c c } 2 & - 8 \ - 2 & 6 \end{array} \right| = - 4$
$$C _ { 13 } = - 4$$
C) $M _ { 13 } = \left| \begin{array} { c c } 3 & - 2 \ - 1 & 3 \end{array} \right| = 7$
$$C _ { 13 } = - 7$$
D) $M _ { 13 } = \left| \begin{array} { c c } 2 & - 8 \ - 2 & 6 \end{array} \right| = - 4$ $C _ { 13 } = 4$
E)
$$\begin{array} { l }
M _ { 13 } = \left| \begin{array} { c c }
3 & - 2 \
- 1 & 3
\end{array} \right| = 7 \
C _ { 13 } = 7
\end{array}$$

Question

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } - 3 & 2 & - 8 \ 3 & - 2 & 6 \ - 1 & 3 & - 6 \end{array} \right]$.
A) $M _ { 13 } = \left| \begin{array} { r r } - 3 & - 8 \ - 1 & - 6 \end{array} \right| = 10$
$$C _ { 13 } = - 10$$
B) $M _ { 13 } = \left| \begin{array} { c c } 2 & - 8 \ - 2 & 6 \end{array} \right| = - 4$
$$C _ { 13 } = - 4$$
C) $M _ { 13 } = \left| \begin{array} { c c } 3 & - 2 \ - 1 & 3 \end{array} \right| = 7$
$$C _ { 13 } = - 7$$
D) $M _ { 13 } = \left| \begin{array} { c c } 2 & - 8 \ - 2 & 6 \end{array} \right| = - 4$ $C _ { 13 } = 4$
E)
$$\begin{array} { l } 
M _ { 13 } = \left| \begin{array} { c c } 
3 & - 2 \
- 1 & 3
\end{array} \right| = 7 \
C _ { 13 } = 7
\end{array}$$

Quizplus · Accepted Answer

The minor $M_{13}$ is found by removing the row and column in which the element $a_{13}$ (the element in the first row and third column) is located, leaving us with the matrix $\left[ \begin{array} { c c } 3 & - 2 \ - 1 & 3 \end{array} \right]$. The determinant of this matrix is $3 \times 3 - (-2) \times (-1) = 9 - 2 = 7$, so $M_{13} = 7$. The cofactor $C_{13}$ is given by $(-1)^{1+3}M_{13} = -1 \times 7 = -7$, but there seems to be a mistake in my explanation regarding the sign of the cofactor. Correcting this, since $C_{13}$ should actually be calculated as $C_{13} = (-1)^{1+3} \times M_{13} = 1 \times 7 = 7$, indicating a mistake in my initial sign calculation. The correct choice, considering the proper calculation of the minor and its cofactor, should reflect $M_{13} = 7$ and $C_{13} = 7$, which is option E.

Find the Minor M13M _ { 13 }M13​ And Its Cofactor C13C _ { 13 }C13​

Find the Minor $M _ { 13 }$ And Its Cofactor $C _ { 13 }$