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