The following matrices are in reduced row echelon form....

The following matrices are in reduced row echelon form. Determine the solution of the corresponding system of linear equations or state that the system is inconsistent. $$ \left[\begin{array}{rrrr|r} 1 & 0 & -8 & 1 & 7 \\ 0 & 1 & 4 & -3 & 2 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] $$

The following matrices are in reduced row echelon form. Determine the solution of the corresponding system of linear equations or state that the system is inconsistent.
$$
\left[\begin{array}{rrrr|r}
1 & 0 & -8 & 1 & 7 \\
0 & 1 & 4 & -3 & 2 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$

Key Concepts

Reduced Row Echelon Form (RREF)

Reduced Row Echelon Form is a form of a matrix used to solve systems of linear equations that guarantees a unique representation of the solution set. In this form, each pivot (leading 1) is the only nonzero entry in its column, and rows with only zeroes are at the bottom of the matrix, making the structure of the solution explicit.

Pivot Variables and Free Variables

In the context of solving systems, pivot variables are those corresponding to the leading ones in the RREF. These are considered the dependent variables. Free variables, on the other hand, correspond to the columns that do not have pivots and can take arbitrary values, allowing the solution set to be expressed in terms of these parameters.

Parametric Form of the General Solution

The general solution of a system in RREF is often expressed in parametric form, where the pivot variables are written in terms of the free variables. This representation clearly shows how the entire solution set is generated, demonstrating both the dependencies and the degrees of freedom within the system.

Consistency of a Linear System

A linear system is consistent if there exists at least one solution, which is typically verified by ensuring that no row of the augmented matrix in the RREF has zeros in all coefficient entries while having a non-zero entry in the augmented part. This check is critical to determine whether the system has any valid solutions or if it is contradictory.

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