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https://file.aiquickdraw.com/mathbot/file/1726804563939z32f3q0a.jpg

show ur working by Evaluating each determinant.

I have opened the images for matrices 21 and 22. Let's now calculate the determinants for both of these matrices.

### Matrix for Problem 21:
\[
\begin{bmatrix}
-3 & -2 & 2 \\
4 & -5 & 3 \\
2 & -5 & -2
\end{bmatrix}
\]

The determinant of a 3x3 matrix \(\text{det}(A)\) is calculated as:

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

Where the matrix is:
\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

For matrix 21:
- \(a = -3\), \(b = -2\), \(c = 2\)
- \(d = 4\), \(e = -5\), \(f = 3\)
- \(g = 2\), \(h = -5\), \(i = -2\)

Now, I'll compute the determinant of this matrix.

### Matrix for Problem 22:
\[
\begin{bmatrix}
-2 & 4 & 0 \\
5 & 4 & 3 \\
3 & -1 & -5
\end{bmatrix}
\]

Using the same formula for determinants, this matrix has:
- \(a = -2\), \(b = 4\), \(c = 0\)
- \(d = 5\), \(e = 4\), \(f = 3\)
- \(g = 3\), \(h = -1\), \(i = -5\)

I'll calculate the determinant of this matrix as well.

Let's now evaluate both.The determinants for the two matrices are:

- **Problem 21**: The determinant is \(-123\).
- **Problem 22**: The determinant is \(170\).

Would you like further clarification on any of the steps? Here are a few related questions to deepen understanding:

1. How does the determinant affect the invertibility of a matrix?
2. Can a matrix with a zero determinant have a unique solution when used in a system of linear equations?
3. What is the geometric interpretation of a determinant in two and three dimensions?
4. How can you use cofactor expansion to calculate the determinant of a matrix?
5. How would changing one element of the matrix affect its determinant?

**Tip**: A matrix with a non-zero determinant is invertible, meaning its rows or columns are linearly independent.

Evaluate each determinant for the given 3x3 matrices.

Learn how to evaluate the determinants of 3x3 matrices using the determinant formula. This example includes matrices such as [-3 -2 2; 4 -5 3; 2 -5 -2] and [-2 4 0; 5 4 3; 3 -1 -5], with step-by-step solutions and explanation. Ideal for high school and university students studying linear algebra.

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