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Let's go through the problems step by step, starting with problem 1 (solving systems of equations using Gaussian elimination).

### Problem 1 (Gaussian Elimination):

**System a:**
\[
\begin{aligned}
2x_1 - 2x_2 + 2x_3 &= -2 \\
x_1 + x_2 - 4x_3 &= 0 \\
3x_1 - x_2 - x_3 &= -1
\end{aligned}
\]

We want to convert this system into row echelon form using Gaussian elimination. The augmented matrix is:

\[
\left[
\begin{array}{ccc|c}
2 & -2 & 2 & -2 \\
1 & 1 & -4 & 0 \\
3 & -1 & -1 & -1
\end{array}
\right]
\]

**Step 1: Pivoting and elimination.**
- First, divide the first row by 2 to simplify it.
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & -1 \\
1 & 1 & -4 & 0 \\
3 & -1 & -1 & -1
\end{array}
\right]
\]

- Now subtract row 1 from row 2 and subtract 3 times row 1 from row 3.
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & -1 \\
0 & 2 & -5 & 1 \\
0 & 2 & -4 & 2
\end{array}
\right]
\]

**Step 2: Eliminate further.**
- Subtract row 2 from row 3 to eliminate the second row in the third row.
\[
\left[
\begin{array}{ccc|c}
1 & -1 & 1 & -1 \\
0 & 2 & -5 & 1 \\
0 & 0 & 1 & 1
\end{array}
\right]
\]

**Step 3: Back substitution.**
- From the third row, we have \(x_3 = 1\).
- Substitute \(x_3 = 1\) into the second row:
\[
2x_2 - 5(1) = 1 \quad \Rightarrow \quad 2x_2 = 6 \quad \Rightarrow \quad x_2 = 3
\]
- Substitute \(x_2 = 3\) and \(x_3 = 1\) into the first row:
\[
x_1 - 3 + 1 = -1 \quad \Rightarrow \quad x_1 = 1
\]

**Solution:**
\[
x_1 = 1, \quad x_2 = 3, \quad x_3 = 1
\]

This system has a unique solution.

---

I'll now proceed with part **b** and **c**, or I can summarize part 2 (inverse matrices) next. Let me know how you'd like to proceed, and if you have any questions so far!

Also, here are some questions to extend the topic:
1. What is the significance of row echelon form in Gaussian elimination?
2. How can Gaussian elimination help in determining if a system has no solution, one solution, or infinitely many solutions?
3. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
4. How do you interpret a row of zeros in an augmented matrix?
5. What steps would you take to solve a homogeneous system using Gaussian elimination?

**Tip**: Always check your row operations carefully to avoid small arithmetic errors that can change the solution!

Solve the following system of linear equations using Gaussian Elimination. Also, find inverse matrices if possible, and solve the system using the inverse matrix where applicable.

Learn how to solve systems of linear equations using Gaussian Elimination and how to find matrix inverses. This guide walks through step-by-step solutions for solving multiple linear systems and applying matrix inverses where applicable. Suitable for undergraduate students in Linear Algebra courses.

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