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Compute the determinant by cofactor expansion

To compute the determinant of the given 4x4 matrix using cofactor expansion, we'll proceed step by step. Here's the matrix extracted from the image:

\[
\begin{pmatrix}
6 & 0 & 0 & 5 \\
1 & 7 & 2 & -5 \\
2 & 3 & 0 & 0 \\
8 & 3 & 1 & 8
\end{pmatrix}
\]

### Step 1: Cofactor expansion along the first row (row 1)
We'll expand along the first row using the cofactor formula:

\[
\text{det}(A) = \sum_{j=1}^{4} (-1)^{1+j} a_{1j} \cdot \text{det}(M_{1j})
\]

Where \( a_{1j} \) are the elements of the first row, and \( M_{1j} \) are the minors obtained by deleting the first row and the \( j \)-th column.

### Step 2: Apply cofactor expansion
We perform the expansion using the elements of the first row \( (6, 0, 0, 5) \):

\[
\text{det}(A) = 6 \cdot \text{det}(M_{11}) - 0 \cdot \text{det}(M_{12}) + 0 \cdot \text{det}(M_{13}) + 5 \cdot \text{det}(M_{14})
\]

Since the second and third terms contain 0, they vanish, leaving us with:

\[
\text{det}(A) = 6 \cdot \text{det}(M_{11}) + 5 \cdot \text{det}(M_{14})
\]

### Step 3: Compute the minors \( M_{11} \) and \( M_{14} \)

#### Minor \( M_{11} \) (delete row 1, column 1):
\[
M_{11} = \begin{pmatrix} 7 & 2 & -5 \\ 3 & 0 & 0 \\ 3 & 1 & 8 \end{pmatrix}
\]

Now, calculate the determinant of this 3x3 matrix using cofactor expansion on the first row:

\[
\text{det}(M_{11}) = 7 \cdot \left| \begin{matrix} 0 & 0 \\ 1 & 8 \end{matrix} \right| - 2 \cdot \left| \begin{matrix} 3 & 0 \\ 3 & 8 \end{matrix} \right| + (-5) \cdot \left| \begin{matrix} 3 & 0 \\ 3 & 1 \end{matrix} \right|
\]

\[
= 7 \cdot (0 \cdot 8 - 0 \cdot 1) - 2 \cdot (3 \cdot 8 - 0 \cdot 3) + (-5) \cdot (3 \cdot 1 - 3 \cdot 0)
\]

\[
= 7 \cdot 0 - 2 \cdot 24 + (-5) \cdot 3 = -48 - 15 = -63
\]

#### Minor \( M_{14} \) (delete row 1, column 4):
\[
M_{14} = \begin{pmatrix} 1 & 7 & 2 \\ 2 & 3 & 0 \\ 8 & 3 & 1 \end{pmatrix}
\]

Similarly, calculate the determinant of this 3x3 matrix:

\[
\text{det}(M_{14}) = 1 \cdot \left| \begin{matrix} 3 & 0 \\ 3 & 1 \end{matrix} \right| - 7 \cdot \left| \begin{matrix} 2 & 0 \\ 8 & 1 \end{matrix} \right| + 2 \cdot \left| \begin{matrix} 2 & 3 \\ 8 & 3 \end{matrix} \right|
\]

\[
= 1 \cdot (3 \cdot 1 - 0 \cdot 3) - 7 \cdot (2 \cdot 1 - 0 \cdot 8) + 2 \cdot (2 \cdot 3 - 8 \cdot 3)
\]

\[
= 1 \cdot 3 - 7 \cdot 2 + 2 \cdot (-18) = 3 - 14 - 36 = -47
\]

### Step 4: Calculate the determinant
Now, substitute these values back into the determinant expression:

\[
\text{det}(A) = 6 \cdot (-63) + 5 \cdot (-47) = -378 - 235 = -613
\]

### Final Answer:
The determinant of the matrix is:

\[
\boxed{-613}
\]

Would you like a further breakdown or clarification?

Here are some follow-up questions to explore:
1. What are the steps involved in cofactor expansion for a general matrix?
2. How do you compute a determinant of a 3x3 matrix using minors?
3. Can the determinant be zero, and what does it imply for the matrix?
4. How does cofactor expansion work for larger matrices?
5. What are alternative methods to compute the determinant?

**Tip:** Cofactor expansion is computationally expensive for large matrices. For efficiency, consider using row or column operations when possible.

Learn how to compute the determinant of a 4x4 matrix using cofactor expansion. This detailed solution covers concepts of linear algebra, determinants, and minors, ideal for high school and college students.

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