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Let's tackle the questions one by one:

### 1. Solve the linear system
\[ x + y = 4 \]
\[ 3x - y = 8 \]

First, add the two equations to eliminate \( y \):
\[ x + y + 3x - y = 4 + 8 \]
\[ 4x = 12 \]
\[ x = 3 \]

Now, substitute \( x = 3 \) into the first equation to find \( y \):
\[ 3 + y = 4 \]
\[ y = 1 \]

So, the solution to the system is:
\[ x = 3, \; y = 1 \]

### 2. Matrix Operations
Given:
\[ A = \begin{pmatrix} 8 & 0 & -3 \\ 1 & -1 & 3 \end{pmatrix}, \]
\[ B = \begin{pmatrix} 2 & 1 & 0 \\ 5 & 9 & 3 \end{pmatrix}, \]
\[ C = \begin{pmatrix} 5 & 2 \\ 6 & -7 \end{pmatrix} \]

**a) Find \( A + B \)**:
\[ A + B = \begin{pmatrix} 8 + 2 & 0 + 1 & -3 + 0 \\ 1 + 5 & -1 + 9 & 3 + 3 \end{pmatrix} = \begin{pmatrix} 10 & 1 & -3 \\ 6 & 8 & 6 \end{pmatrix} \]

**b) Find \( CA \)**:
\[ C = \begin{pmatrix} 5 & 2 \\ 6 & -7 \end{pmatrix} \]
\[ A = \begin{pmatrix} 8 & 0 & -3 \\ 1 & -1 & 3 \end{pmatrix} \]
Since the matrices \( C \) and \( A \) are not conformable for multiplication (dimensions do not match), this operation is not possible.

### 3. Find the inverse of the matrix
\[ \begin{pmatrix} 3 & 1 \\ 8 & 3 \end{pmatrix} \]

The inverse of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For the matrix \( \begin{pmatrix} 3 & 1 \\ 8 & 3 \end{pmatrix} \):
\[ a = 3, \; b = 1, \; c = 8, \; d = 3 \]
\[ ad - bc = 3 \cdot 3 - 1 \cdot 8 = 9 - 8 = 1 \]

Thus, the inverse is:
\[ \begin{pmatrix} 3 & -1 \\ -8 & 3 \end{pmatrix} \]

### 4. Find the determinant of the matrix
**a)**
\[ \begin{pmatrix} 9 & 3 \\ 8 & 3 \end{pmatrix} \]
\[ \text{Det} = (9 \cdot 3) - (3 \cdot 8) = 27 - 24 = 3 \]

**b)**
\[ \begin{pmatrix} 1 & 0 & 0 \\ -1 & 0 & 1 \\ 2 & 4 & -3 \end{pmatrix} \]
The determinant of a 3x3 matrix \( \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is:
\[ a(ei - fh) - b(di - fg) + c(dh - eg) \]

For the given matrix:
\[ a = 1, \; b = 0, \; c = 0, \; d = -1, \; e = 0, \; f = 1, \; g = 2, \; h = 4, \; i = -3 \]
\[ \text{Det} = 1(0 \cdot -3 - 1 \cdot 4) - 0(-1 \cdot -3 - 1 \cdot 2) + 0(-1 \cdot 4 - 0 \cdot 2) \]
\[ \text{Det} = 1(0 - 4) = -4 \]

The determinant is \(-4\).

Would you like more details on any of these solutions, or do you have any other questions? Here are some follow-up questions you might be interested in:

1. Can you explain how to solve a system of equations using matrix methods?
2. What are some properties of determinants in matrix algebra?
3. How do you find the inverse of a larger matrix?
4. What is the significance of a matrix determinant being zero?
5. Can you show the steps for multiplying two matrices?

**Tip

Learn how to solve linear equations and perform matrix operations step-by-step. Includes solutions for linear systems, matrix addition, and determinant calculations. Suitable for advanced high school students.

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