Question

Given the matrix:

$\begin{pmatrix} 3 & 5 & 2 \\ 1 & 4 & 6 \\ 7 & 0 & 8 \end{pmatrix}$

Find the minor corresponding to the element in the 2nd row and 3rd column, and then determine its cofactor.

Answer 1

Minor corresponding to element at (2,3): -35

Cofactor corresponding to element at (2,3): 35

Answer 2

Solution:

1. Finding the Minor:
   For element $a_{23} = 6$ (2nd row, 3rd column), remove the 2nd row and 3rd column to obtain the $2 \times 2$ matrix:

   $$\begin{pmatrix} 3 & 5 \\ 7 & 0 \end{pmatrix}$$

   Its determinant (the minor) is:

   $$M_{23} = 3 \times 0 - 7 \times 5 = -35.$$

2. Finding the Cofactor:
   The cofactor is given by:

   $$C_{23} = (-1)^{2+3} \times M_{23} = (-1)^5 \times (-35) = -(-35) = 35.$$

Explanation (Minimal Core):
Remove row 2 and column 3, compute the determinant $\det\begin{pmatrix} 3 & 5 \\ 7 & 0 \end{pmatrix} = -35$ for the minor, and then multiply by $(-1)^{2+3} = -1$ to get the cofactor $35$.