The image you uploaded contains a mathematical problem with several assertions regarding linear algebra concepts, specifically dealing with linear systems and their solutions. Let’s break them down one by one:

Problem 3: Decide if the following assertions are always true or sometimes false. Justify your answers.

1. (i) If the augmented matrix of a linear system of equations has a pivot in each row, then the system is consistent (i.e., has a solution).

   • Analysis: This statement is true. If every row has a pivot in the augmented matrix, this implies that there is no row of the form $[0\ 0\ ...\ 0\ |\ b]$ with $b \neq 0$, which would indicate inconsistency. Hence, the system must have at least one solution.

2. (ii) If a linear system of equations is inconsistent, then the coefficient parts of its RREF augmented matrix cannot have a pivot in each column.

   • Analysis: This statement is false. An inconsistent system can still have a pivot in each column of the coefficient matrix but may have a row like $[0\ 0\ ...\ 0\ |\ b]$, where $b \neq 0$, leading to inconsistency.

3. (iii) If an echelon form of a linear system of equations has fewer pivots than the system has variables, then the system has infinitely many solutions.

   • Analysis: This statement is true. If there are fewer pivots than variables, the system has free variables, implying that the system has an infinite number of solutions (unless inconsistent).

4. (iv) If a non-homogeneous linear system of equations has exactly one solution, then the associated linear system of equations has no non-trivial solution.

   • Analysis: This statement is true. If a non-homogeneous system has exactly one solution, the corresponding homogeneous system (Ax = 0) will only have the trivial solution, which means no non-trivial solutions exist.

5. (v) If $\begin{bmatrix} 1 \\ 3 \\ -3 \end{bmatrix} \in \mathbb{R}^3$ is a solution of a non-homogeneous linear system of equations $Ax = b, b \neq 0$, and if $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} s + 3t \\ -t \\ s - t \end{bmatrix} \in \mathbb{R}^3$ is the general solution of the associated homogeneous linear system of equations $Ax = 0$, then $\begin{bmatrix} 1 \\ 3 \\ -3 \end{bmatrix} + \begin{bmatrix} s + 3t \\ -t \\ s - t \end{bmatrix} \in \mathbb{R}^3$ is the general solution of the non-homogeneous linear system.

   • Analysis: This statement is true. The general solution of the non-homogeneous system is the particular solution (given as $\begin{bmatrix} 1 \\ 3 \\ -3 \end{bmatrix}$) plus the general solution of the homogeneous system. Therefore, adding the two gives the general solution of the non-homogeneous system.

Would you like further explanations on any of these points?

Here are some related questions to further explore this topic:

1. How can you determine if a system of linear equations is consistent from the augmented matrix?
2. What is the role of free variables in determining the number of solutions to a linear system?
3. How does the rank of a matrix relate to the number of solutions of a system of linear equations?
4. What is the difference between a homogeneous and a non-homogeneous system of linear equations?
5. How do pivots in reduced row echelon form (RREF) influence the solution set of a system?

Tip: When solving linear systems, always pay attention to the augmented matrix structure, as it provides essential information about the consistency and nature of the solutions (unique, infinite, or none).