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.