https://file.aiquickdraw.com/mathbot/file/1726504173822tpbsb5ca.jpg

Let's evaluate the determinant for each of the matrices provided in the image.

1. For the 2x2 matrix:
\[
\begin{bmatrix}
3 & 2 \\
4 & -3
\end{bmatrix}
\]
The determinant is calculated as follows:
\[
\text{det}(A) = (3 \times -3) - (2 \times 4) = -9 - 8 = -17
\]

2. For the 3x3 matrix:
\[
\begin{bmatrix}
1 & 5 & -4 \\
5 & 2 & -3 \\
-3 & -2 & 3
\end{bmatrix}
\]
The determinant for a 3x3 matrix, \(A\), can be calculated using the rule of Sarrus or cofactor expansion. We will use the rule of Sarrus here:
\[
\text{det}(A) = (1 \times 2 \times 3) + (5 \times -3 \times -3) + (-4 \times 5 \times -2) - (-4 \times 2 \times -3) - (1 \times -3 \times -2) - (5 \times 5 \times 3)
\]
Calculating this gives:
\[
\text{det}(A) = (6) + (45) + (40) - (24) - (6) - (75)
\]
\[
= 91 - 105 = -14
\]

So the determinants of the matrices are:
- For the 2x2 matrix: \(-17\)
- For the 3x3 matrix: \(-14\)

Would you like more details on any of these steps, or do you have any other questions?

Here are some additional related questions you might consider:
1. How does changing an element in a matrix affect its determinant?
2. What is the geometric interpretation of a determinant?
3. How would adding a multiple of one row to another row affect the determinant?
4. Can you explain how the determinant relates to the invertibility of a matrix?
5. What happens to the determinant when you transpose a matrix?

**Tip:** Remember, the determinant of a matrix is zero if and only if the matrix is singular, meaning it does not have an inverse.

Evaluate the determinant of the given matrices: 5) A 2x2 matrix [3, 2; 4, -3], 6) A 3x3 matrix [1, 5, -4; 5, 2, -3; -3, -2, 3]

Learn how to calculate the determinants of a 2x2 matrix [3, 2; 4, -3] and a 3x3 matrix [1, 5, -4; 5, 2, -3; -3, -2, 3] using key linear algebra formulas like Sarrus' rule. Detailed step-by-step solutions are provided, ideal for high school students in Grades 10-12.

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