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An augmented matrix is a crucial tool in representing linear systems. It combines both the coefficient matrix of the variables and the constants from equations into a single matrix.
This form helps us efficiently perform manipulations to solve systems of equations. The matrix is structured as follows:

• The left part consists of the coefficients of each variable in the system of equations.
• The right part (after a vertical line) represents the constant terms.

To illustrate, consider the augmented matrix \[\begin{bmatrix}0 & 0 & 1 & | & 2 \0 & 1 & 3 & | & 1 \1 & 0 & 1 & | & 1\end{bmatrix}\]The expressions before the vertical line are the coefficients of variables \(x\), \(y\), and \(z\), and the numbers to the right are the constants from each equation in the system. This makes it easier to apply operations to solve the equations efficiently.
By transforming this matrix, we can find solutions to the system easily, which is often the goal when using augmented matrices.

In the process of solving linear equations using matrices, transforming an augmented matrix into row echelon form (REF) is a fundamental step. This form simplifies systems to make solution discovery easy.
In row echelon form, a matrix typically features:

• Each row having a leading '1', known as a pivot, that shifts to the right as you move from the first row downward.
• Any rows with all zero elements, which are located at the bottom of the matrix.
• Each leading '1' is to the right of the leading '1' in the previous row.

For example, consider the matrix given in the exercise:\[\begin{bmatrix}0 & 0 & 1 \0 & 1 & 3 \1 & 0 & 1\end{bmatrix}\]This matrix already approximates row echelon form because it features the necessary downward shift of leading '1's. By achieving this form, it becomes simpler to determine the status and nature of solutions to the system of equations.This form reveals if a system is consistent or inconsistent by visually showing dependencies between rows.

Pivot columns are an essential concept in linear algebra, particularly when transforming a matrix into row echelon form. Identifying these columns helps determine the rank of the matrix and the nature of solutions for a system.
A pivot column is one that contains a leading '1' after the matrix has been transformed into row echelon form. Characteristics of pivot columns include:

• Every pivot column corresponds to a basic variable in the system.
• The number of pivot columns indicates the number of independent equations in the system.
• A system with a pivot in each variable column typically has a unique solution.

In the original exercise, the matrix was assessed for pivot columns:\[\begin{bmatrix}0 & 0 & 1 & | & 2 \0 & 1 & 3 & | & 1 \1 & 0 & 1 & | & 1\end{bmatrix}\]Here, columns 1, 2, and 3 are pivot columns representing \(x\), \(y\), and \(z\) respectively. This classification facilitated the determination that the system had a unique solution since each variable aligns with a pivot point.