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 $k$.

Where would you like to start? Would you prefer I begin solving the first question, or is there a specific section you'd like to explore?