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The image shows a mathematical problem related to matrix operations and solving systems of linear equations. I’ll extract the relevant parts and provide a solution:

### 1(a) Matrix Multiplication

You are asked to find the result of matrix multiplication:

\[
\begin{bmatrix}
1 & -2 \\
-3 & 4
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
5 \\
-7
\end{bmatrix}
\]

### 1(b) Solve the System

The system of linear equations given in matrix form can be written as:
- \( 1 \cdot x_1 - 2 \cdot x_2 = 5 \)
- \( -3 \cdot x_1 + 4 \cdot x_2 = -7 \)

You need to solve this system for \(x_1\) and \(x_2\).

### 2(c) Finding Determinant and Solving

In the third part, you are given the matrix:

\[
\begin{bmatrix}
2 & -3 & 0 \\
1 & 4 & -1 \\
5 & -2 & -1
\end{bmatrix}
\]

And you need to find the determinant and then use it to solve a system involving the matrix.

### Solutions:

#### 1(a) Matrix Multiplication
The system can be written as:

\[
\begin{aligned}
1 \cdot x_1 - 2 \cdot x_2 &= 5 \\
-3 \cdot x_1 + 4 \cdot x_2 &= -7
\end{aligned}
\]

#### 1(b) Solve for \(x_1\) and \(x_2\):
Let's solve it using substitution or elimination method.

1. First equation: \( x_1 - 2x_2 = 5 \)
2. Second equation: \( -3x_1 + 4x_2 = -7 \)

Multiply the first equation by 3:

\[
3x_1 - 6x_2 = 15
\]

Now add this to the second equation:

\[
(3x_1 - 6x_2) + (-3x_1 + 4x_2) = 15 - 7
\]

Simplifying:

\[
-2x_2 = 8 \implies x_2 = -4
\]

Substitute \(x_2 = -4\) into the first equation:

\[
x_1 - 2(-4) = 5 \implies x_1 + 8 = 5 \implies x_1 = -3
\]

So, \( x_1 = -3 \) and \( x_2 = -4 \).

#### 2(c) Determinant Calculation:
Calculate the determinant of the given 3x3 matrix:

\[
\text{det} = 2 \times \text{det} 
\begin{bmatrix}
4 & -1 \\
-2 & -1
\end{bmatrix}
- (-3) \times \text{det}
\begin{bmatrix}
1 & -1 \\
5 & -1
\end{bmatrix}
+ 0 \times \text{det} 
\begin{bmatrix}
1 & 4 \\
5 & -2
\end{bmatrix}
\]

\[
\text{det} = 2 \times (4 \times -1 - (-1) \times -2) - (-3) \times (1 \times -1 - (-1) \times 5)
\]

\[
= 2 \times (-4 + 2) + 3 \times (-1 + 5)
\]

\[
= 2 \times (-2) + 3 \times 4
\]

\[
= -4 + 12 = 8
\]

The determinant of the matrix is 8.

---

### Additional Help

If you want further detailed explanations or if you have any other questions, feel free to ask!

Here are some follow-up questions for further exploration:

1. What is the inverse of the matrix given in 1(a)?
2. How would the solutions change if the determinant in 2(c) was zero?
3. Can you generalize the steps to solve any 2x2 matrix system of linear equations?
4. How does matrix multiplication differ from scalar multiplication?
5. How can you verify the solution obtained for \(x_1\) and \(x_2\)?

**Tip:** When solving linear equations using matrices, always check if the matrix is invertible (non-zero determinant) to ensure a unique solution exists.

Explore matrix operations and linear equations with detailed step-by-step solutions. Learn how to multiply matrices, solve systems of linear equations, and calculate determinants effectively. Suitable for advanced high school students and anyone interested in linear algebra concepts.

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