Let's solve each of the three questions based on the image you've uploaded.

Question 1:

You are asked to find the determinant of matrix $A$, which is given as:

$A = \begin{pmatrix} 2 & -2 \\ 1 & 5 \end{pmatrix}$

The determinant of a 2x2 matrix is calculated using the formula:

$\text{det}(A) = (a \times d) - (b \times c)$

For matrix $A$:

$a = 2, \, b = -2, \, c = 1, \, d = 5$

$\text{det}(A) = (2 \times 5) - (-2 \times 1) = 10 + 2 = 12$

So, the determinant of $A$ is 12. The correct answer is c).

Question 2:

For the second question, you are given the matrix $A$:

$A = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$

The determinant of this 2x2 matrix is also calculated as:

$\text{det}(A) = (e \times h) - (f \times g)$

This matches the formula given as option a), so the correct answer is a).

Question 3:

You are given two matrices $A$ and $B$ and asked to find the sum of their determinants:

$A = \begin{pmatrix} -2 & 7 \\ -2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 5 \\ 1 & -3 \end{pmatrix}$

Step 1: Determinant of $A$

Using the formula for a 2x2 determinant:

$\text{det}(A) = (-2 \times 3) - (7 \times -2) = -6 + 14 = 8$

Step 2: Determinant of $B$

$\text{det}(B) = (4 \times -3) - (5 \times 1) = -12 - 5 = -17$

Step 3: Sum of Determinants

$\text{det}(A) + \text{det}(B) = 8 + (-17) = -9$

Thus, the correct answer is d).

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How do we compute the determinant of a 3x3 matrix?
2. What is the significance of a matrix having a zero determinant?
3. Can you explain how the determinant is used in solving systems of equations?
4. How do properties of determinants change if we scale or transpose a matrix?
5. What is the geometric interpretation of a matrix determinant?

Tip: The determinant is crucial in determining if a matrix is invertible. If the determinant is zero, the matrix does not have an inverse.