Step 1: Identify the position and sign: The element -4 is located in the second row ($i=2$) and the first column ($j=1$).
The formula for a co-factor $C_{ij}$ is:
$$C_{ij} = (-1)^{i+j} \cdot M_{ij}$$
Where $M_{ij}$ is the minor of the element. For our position (2,1):
$$\text{Sign} = (-1)^{2+1} = (-1)^3 = -1$$
Step 2: Find the Minor ($M_{21}$): The minor is found by deleting the row and column containing -4 (the 2nd row and 1st column). The remaining elements form a $2 \times 2$ determinant:
$$M_{21} = \begin{vmatrix} 2 & 3 \\ -7 & 9 \end{vmatrix}$$
Calculating the value:
$$M_{21} = (2 \times 9) - (3 \times -7)$$
$$M_{21} = 18 - (-21) = 18 + 21 = 39$$
Step 3: Calculate the Co-factor: Combine the sign and the minor:
$$C_{21} = (-1) \times 39 = -39$$
Thus, the co-factor of -4 in the given determinant is -39.