https://file.aiquickdraw.com/mathbot/file/1727448812354xrdmwij3.jpg

The image displays two matrices \( A \) and \( B \), and the task is to find their determinants.

1. **Matrix \( A \)** is a \( 2 \times 2 \) matrix:
   \[
   A = \begin{bmatrix} 7 & -2 \\ 4 & 3 \end{bmatrix}
   \]
   The determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is computed as:
   \[
   \text{det}(A) = ad - bc
   \]

2. **Matrix \( B \)** is a \( 3 \times 3 \) matrix:
   \[
   B = \begin{bmatrix} 3 & 0 & -1 \\ 2 & 5 & 4 \\ -1 & -1 & 6 \end{bmatrix}
   \]
   The determinant of a \( 3 \times 3 \) matrix is computed using cofactor expansion. Let's compute both determinants.The determinants of the matrices are:

- \( \text{det}(A) = 29 \)
- \( \text{det}(B) = 99 \)

Would you like a more detailed explanation of the steps involved, or do you have any other questions?

Here are 5 related questions to deepen your understanding:
1. How is the cofactor expansion used to compute the determinant of larger matrices?
2. What is the significance of the determinant of a matrix in linear algebra?
3. Can a matrix with a zero determinant have an inverse?
4. How would the determinant change if one row of the matrix is multiplied by a scalar?
5. How are determinants applied in solving systems of linear equations?

**Tip:** The determinant of a matrix can also be used to determine if a set of vectors is linearly independent.

Find the determinant of matrices A and B. A is a 2x2 matrix and B is a 3x3 matrix.

Learn how to find the determinant of matrices A and B, where A is a 2x2 matrix and B is a 3x3 matrix. This solution covers essential linear algebra concepts like the 2x2 determinant formula and the cofactor expansion for 3x3 matrices. Suitable for Grades 11-12 students.

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