)	Find the determinant of the following matrix.

A = 	
−3	0	2	0
2	2	0	−3
1	0	−4	−4
1	0	−4	0

To find the determinant of the given \( 4 \times 4 \) matrix \( A \):

\[
A = \begin{bmatrix}
-3 & 0 & 2 & 0 \\
2 & 2 & 0 & -3 \\
1 & 0 & -4 & -4 \\
1 & 0 & -4 & 0
\end{bmatrix}
\]

### Step-by-Step Calculation

1. **Choose the first row for expansion:**
   
\[
\text{det}(A) = (-3) \cdot C_{11} + 0 \cdot C_{12} + 2 \cdot C_{13} + 0 \cdot C_{14}
\]

Here, \( C_{ij} \) represents the cofactor of the element in the \( i \)-th row and \( j \)-th column.

2. **Calculate the cofactors:**
   
   - **Cofactor \( C_{11} \):**
   
     \[
     C_{11} = \begin{vmatrix}
     2 & 0 & -3 \\
     0 & -4 & -4 \\
     0 & -4 & 0
     \end{vmatrix}
     \]
     
     Compute this determinant:
     
     \[
     C_{11} = 2 \cdot ((-4) \cdot 0 - (-4) \cdot (-4)) = 2 \cdot (0 - 16) = -32
     \]
   
   - **Cofactor \( C_{13} \):**
   
     \[
     C_{13} = \begin{vmatrix}
     2 & 2 & -3 \\
     1 & 0 & -4 \\
     1 & 0 & 0
     \end{vmatrix}
     \]
     
     Compute this determinant:
     
     \[
     C_{13} = 2 \cdot (0 \cdot 0 - (-4) \cdot 0) - 2 \cdot (1 \cdot 0 - (-4) \cdot 1) + (-3) \cdot (1 \cdot 0 - 0 \cdot 1)
     \]
     
     Simplify:
     
     \[
     C_{13} = 0 + 8 + 0 = 8
     \]

3. **Substitute the cofactors back into the determinant expression:**
   
\[
\text{det}(A) = (-3)(-32) + 2 \cdot 8
\]
   
Simplify:

\[
\text{det}(A) = 96 + 16 = 112
\]

### Final Answer:
\[
\text{det}(A) = 112
\]

Would you like more details on the process or have any questions?

### Related Questions
1. How do you compute a determinant for a \(3 \times 3\) matrix?
2. What is the significance of the determinant being zero?
3. Can you find the inverse of a \(4 \times 4\) matrix?
4. How do you use cofactor expansion along different rows?
5. What are some applications of determinants in geometry?

### Tip
Always choose a row or column with the most zeros for cofactor expansion to simplify the calculations.

Find the determinant of the following matrix.

A =
−3	0	2	0
2	2	0	−3
1	0	−4	−4
1	0	−4	0

Calculate the determinant of a 4x4 matrix using cofactor expansion. This step-by-step solution includes computing cofactors and applying the cofactor expansion theorem, ideal for Grades 11-12 students learning advanced matrix algebra.

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