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The image asks us to find \( C_{31} \), which is a cofactor of the matrix \( A \) at the position corresponding to row 3, column 1. The matrix \( A \) is:

\[
A = \begin{bmatrix}
-1 & -9 & 9 \\
-3 & -5 & 8 \\
-2 & 4 & 2
\end{bmatrix}
\]

### Steps to Find the Cofactor \( C_{31} \):

1. **Delete the 3rd row and 1st column** to find the 2x2 minor matrix:
   \[
   M_{31} = \begin{bmatrix}
   -9 & 9 \\
   -5 & 8
   \end{bmatrix}
   \]

2. **Calculate the determinant** of the minor matrix:
   \[
   \det(M_{31}) = (-9)(8) - (9)(-5) = -72 + 45 = -27
   \]

3. **Apply the cofactor formula**:
   \[
   C_{31} = (-1)^{3+1} \cdot \det(M_{31}) = (-1)^4 \cdot (-27) = 1 \cdot (-27) = -27
   \]

Thus, \( C_{31} = -27 \).

Would you like further details on cofactor calculations or any other questions?

### Follow-up Questions:
1. How is a minor matrix related to the determinant?
2. Why do we apply the sign \( (-1)^{i+j} \) in the cofactor calculation?
3. Can you explain how to calculate the determinant of a 2x2 matrix?
4. How would you find the cofactor for another element of the matrix?
5. What role do cofactors play in calculating the determinant of larger matrices?

#### Tip:
When calculating determinants of larger matrices, remember to choose the row or column with the most zeros to simplify cofactor expansion.

Learn how to calculate the cofactor C_{31} of matrix A using minor matrices and determinants. This problem explores linear algebra concepts such as cofactor expansion and determinant calculations. Suitable for students in Grades 10-12.

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