Question: Given matrix K =
$$\begin{pmatrix} y & 1 & 3 \\ 1 & z & x \\ -3 & 2 & 2 \end{pmatrix}$$.
Find minor, $m_{13}$ and cofactor, $c_{23}$.
1. **State the problem:**
We are given the matrix
$$K = \begin{pmatrix} y & 1 & 3 \\ 1 & z & x \\ -3 & 2 & 2 \end{pmatrix}$$
and need to find the minor $m_{13}$ and the cofactor $c_{23}$.
2. **Recall definitions:**
- The minor $m_{ij}$ is the determinant of the submatrix formed by deleting the $i$-th row and $j$-th column.
- The cofactor $c_{ij} = (-1)^{i+j} m_{ij}$.
3. **Find minor $m_{13}$:**
- Delete row 1 and column 3:
$$\begin{pmatrix} 1 & z \\ -3 & 2 \end{pmatrix}$$
- Calculate determinant:
$$m_{13} = (1)(2) - (-3)(z) = 2 + 3z$$
4. **Find cofactor $c_{23}$:**
- Delete row 2 and column 3:
$$\begin{pmatrix} y & 1 \\ -3 & 2 \end{pmatrix}$$
- Calculate minor:
$$m_{23} = y \cdot 2 - (-3) \cdot 1 = 2y + 3$$
- Calculate cofactor:
$$c_{23} = (-1)^{2+3} m_{23} = (-1)^5 (2y + 3) = - (2y + 3)$$
**Final answers:**
$$m_{13} = 2 + 3z$$
$$c_{23} = - (2y + 3)$$