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I see that you've uploaded an image containing a variety of matrix-based and linear algebra problems. Let's break these down and work through each section.

### Problem Overview

1. **Find the Rank of Matrices (Echelon Form & Normal Form)**
   - There are three matrices for which the rank needs to be determined using two methods: *Echelon Form* and *Normal Form*.

2. **Inverse of Matrices using Gauss-Jordan Method**
   - Three matrices need to be inverted using the *Gauss-Jordan* method.

3. **Find Non-Singular Matrices P and Q such that \( PAQ \) is in Normal Form**
   - Three matrices are provided, and the task is to find matrices \( P \) and \( Q \) that transform the given matrices into normal form.

4. **Test for Consistency and Solve Linear Systems**
   - Multiple linear systems are presented for consistency checks and solution finding.

5. **Investigate Values of \( \mu \) and \( \lambda \) for Specific Systems**
   - The goal is to investigate for what values of \( \mu \) and \( \lambda \) the given systems have no solution, unique solutions, or an infinite number of solutions.

6. **Values of \( k \) for Consistent Solutions**
   - Find values of \( k \) such that given systems have consistent solutions.

7. **Determine Non-Trivial Solutions for Specific Systems**
   - The objective is to find non-trivial solutions for certain systems of equations by determining \( k \).

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The problem includes finding the rank of matrices, inverting matrices using the Gauss-Jordan method, finding non-singular matrices P and Q such that PAQ is in normal form, testing for consistency and solving linear systems, investigating values of mu and lambda for specific systems, and determining values of k for consistent solutions in given systems of equations.

Explore how to solve matrix problems, including finding the rank using echelon form, inverting matrices via Gauss-Jordan method, transforming matrices to normal form, and testing linear system consistency. This guide covers key topics in linear algebra suitable for university-level students.

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