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The problem is asking to find the determinant of the matrix using expansion by cofactors. The matrix given is:

\[
\begin{bmatrix}
3 & 2 & 0 & 0 & 5 \\
0 & 1 & 3 & 4 & 6 \\
0 & 0 & 4 & 2 & 2 \\
0 & 0 & 6 & 5 & 1 \\
0 & 0 & 0 & 0 & 2
\end{bmatrix}
\]

### Step 1: Recognize Zero Elements
The matrix has several zero entries, which simplifies the cofactor expansion. The most convenient row to expand along is the 5th row, as it contains many zeros.

### Step 2: Cofactor Expansion Along the 5th Row
Since the last row is mostly zeros, only the element in the last column contributes to the determinant. The cofactor expansion for the determinant along the 5th row is:

\[
\text{Determinant} = 2 \times \text{det}\left(\begin{bmatrix}
3 & 2 & 0 & 0 \\
0 & 1 & 3 & 4 \\
0 & 0 & 4 & 2 \\
0 & 0 & 6 & 5
\end{bmatrix}\right)
\]

### Step 3: Expand the Reduced 4x4 Matrix
Next, find the determinant of the 4x4 submatrix:

\[
\begin{bmatrix}
3 & 2 & 0 & 0 \\
0 & 1 & 3 & 4 \\
0 & 0 & 4 & 2 \\
0 & 0 & 6 & 5
\end{bmatrix}
\]

We can expand this along the first row:

\[
\text{Determinant} = 3 \times \text{det}\left(\begin{bmatrix}
1 & 3 & 4 \\
0 & 4 & 2 \\
0 & 6 & 5
\end{bmatrix}\right)
\]

### Step 4: Expand the 3x3 Matrix
Now, calculate the determinant of the 3x3 matrix:

\[
\begin{bmatrix}
1 & 3 & 4 \\
0 & 4 & 2 \\
0 & 6 & 5
\end{bmatrix}
\]

Expanding along the first row:

\[
\text{Determinant} = 1 \times \text{det}\left(\begin{bmatrix}
4 & 2 \\
6 & 5
\end{bmatrix}\right)
\]

The determinant of this 2x2 matrix is:

\[
\text{det} \begin{bmatrix}
4 & 2 \\
6 & 5
\end{bmatrix} = (4)(5) - (2)(6) = 20 - 12 = 8
\]

Thus, the determinant of the 3x3 matrix is:

\[
1 \times 8 = 8
\]

### Step 5: Final Determinant Calculation
Now, going back up the chain of expansions:

\[
\text{Determinant of the 4x4 matrix} = 3 \times 8 = 24
\]

Finally, the determinant of the original 5x5 matrix is:

\[
\text{Determinant} = 2 \times 24 = 48
\]

### Conclusion:
The correct determinant of the matrix is **48**, not 240.

Do you want more details or have any questions? Here are 5 related questions:

1. How does cofactor expansion work for larger matrices?
2. Why is choosing a row or column with zeros useful in cofactor expansion?
3. What would happen if you expanded along a different row?
4. How does the determinant help in solving systems of linear equations?
5. Can determinants be used to compute the inverse of a matrix?

**Tip:** When performing cofactor expansion, always try to pick a row or column with the most zeros to simplify calculations.

Use expansion by cofactors to find the determinant of the matrix: 
[3 2 0 0 5] 
[0 1 3 4 6] 
[0 0 4 2 2] 
[0 0 6 5 1] 
[0 0 0 0 2]

Learn how to find the determinant of a 5x5 matrix using cofactor expansion. This example shows a step-by-step solution, simplifying the matrix by expanding along rows with zeros, and calculates submatrices leading to the final result. Suitable for college-level linear algebra courses.

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