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Find the determinant for the following matrix. 

The matrix given is:

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

To find the determinant of a \(3 \times 3\) matrix, we use the cofactor expansion formula:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For the matrix:

\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

In this case:
- \(a = 2\), \(b = 0\), \(c = -4\)
- \(d = -5\), \(e = 3\), \(f = -2\)
- \(g = -1\), \(h = -1\), \(i = 2\)

Now, let's compute the determinant:

\[
\text{det}(A) = 2 \left( (3)(2) - (-2)(-1) \right) - 0 + (-4) \left( (-5)(-1) - (3)(-1) \right)
\]

Simplifying each term:

\[
\text{det}(A) = 2 \left( 6 - 2 \right) + (-4) \left( 5 + 3 \right)
\]
\[
\text{det}(A) = 2(4) + (-4)(8)
\]
\[
\text{det}(A) = 8 - 32
\]
\[
\text{det}(A) = -24
\]

So, the determinant of the matrix is:

\[
\boxed{-24}
\]

Would you like further clarification or more details on any part of this? 

Here are 5 related questions:
1. What is the determinant of a \(2 \times 2\) matrix?
2. How does the cofactor expansion method generalize to larger matrices?
3. Can the determinant be used to check if a matrix is invertible?
4. What is the geometric interpretation of the determinant?
5. How does row or column operations affect the determinant?

**Tip**: The determinant of a matrix gives information about its invertibility—if the determinant is zero, the matrix is singular and non-invertible.

Find the determinant for the following matrix.

Learn how to find the determinant of a 3x3 matrix using the cofactor expansion method. The solution involves calculating the determinant of the matrix with elements 2, 0, -4, -5, 3, -2, -1, -1, 2. Suitable for students in Grades 10-12.

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