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To solve the determinant of a \( 3 \times 3 \) matrix using cofactor expansion, you mentioned focusing on the second column (\( 0, 0, 7 \)) for the first step. The matrix is:

\[
\begin{vmatrix}
4 & 0 & 5 \\
-1 & 0 & 6 \\
5 & 7 & -5
\end{vmatrix}
\]

We'll use the cofactor expansion along the second column (the one with entries \( 0, 0, 7 \)):

### Step-by-step:

1. For cofactor expansion, we need to calculate the minors for the second column:
   \[
   \text{determinant} = 0 \cdot C_{1,2} - 0 \cdot C_{2,2} + 7 \cdot C_{3,2}
   \]
   The two terms involving \( 0 \cdot \text{Cofactor} \) will disappear, so we only need to calculate the third cofactor \( 7 \cdot C_{3,2} \).

2. The minor for \( C_{3,2} \) (the cofactor corresponding to the position of \( 7 \)) involves deleting the third row and second column, leaving us with the submatrix:
   \[
   \begin{vmatrix}
   4 & 5 \\
   -1 & 6
   \end{vmatrix}
   \]

3. Now, compute the determinant of this \( 2 \times 2 \) submatrix:
   \[
   \text{determinant} = (4)(6) - (-1)(5) = 24 + 5 = 29
   \]

4. Multiply this by the cofactor \( 7 \):
   \[
   7 \times 29 = 203
   \]

### Final Answer:
The determinant of the given \( 3 \times 3 \) matrix is \( 203 \).

Would you like further details on any step or have additional questions?

Here are some follow-up questions:

1. How would the calculation differ if you expanded along a different column or row?
2. Why do the terms involving zeros drop out in cofactor expansion?
3. What are the properties of the determinant of a matrix that help in solving larger matrices?
4. Could you explain why the determinant of a matrix provides insight into the invertibility of the matrix?
5. What are the geometric interpretations of a matrix's determinant?

**Tip:** Cofactor expansion can be simplified by choosing rows or columns with the most zeros, reducing the number of necessary calculations.

Calculate the 3x3 determinant of the matrix: 
| 4  0  5 |
| -1  0  6 |
| 5  7 -5 |

Learn how to calculate the determinant of a 3x3 matrix using cofactor expansion. This example focuses on the matrix | 4 0 5 |, | -1 0 6 |, | 5 7 -5 | and demonstrates key linear algebra concepts suitable for Grade 10-12 students.

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