Consistency

Consistency refers to whether a Linear System has solutions.

Theorem Of Consistency §

A system of linear equations has zero solutions, one solution, or infinitely many solutions. There are no other possibilities.

• If the system has no solutions, it is inconsistent.
• Else, if it has one or infinite solutions, it is consistent.

Every homogenous system is consistent, as there is always at least one solution, by setting all coefficients to 0:

$$\{\begin{array}{l}x+y=z\\ z-y=4x\end{array}\text{ can be solved with x = 0, y = 0, z = 0 giving }\{\begin{array}{l}0=0\\ 0=0\end{array}$$

Theorem of Inconsistency in Matrix Form §

A Linear System is inconsistent if and only if there is at least one row in the row-echelon or reduced row-echelon matrix having all entries equal to zero, except for a non-zero in the final entry

In other words, every row in the coefficient matrix of the row-echelon form must contain a leading entry.

For example

$$\left[\begin{array}{rrr}1& 0& 3\\ 0& 0& 2\end{array}\right]\text{ is incosistent due to }{R}_2$$

If we use back substitution interpret the matrix back into equations, we get:

$$\{\begin{array}{l}x=3\\ 0=2\end{array}$$

which is a contradiction, hence the system has no solutions.

Theorem of Consistency in Coefficient-Augmented Matrix Form §

Suppose we have a consistent linear system with $n$ variables ($n$ columns)

• If the row-reduced augmented matrix has exactly $n$ non-zero rows, then the system has a unique solution.
• If the row-reduced augmented matrix has $<n$ non-zero rows, then the system has infinitely many solutions.
  • If $r$ is the number of non-zero rows in the row-echelon form, then $n-r$ parameters are needed to specify the solution set.

$r$ is the rank of the matrix.

Theorem of Consistency from Rank §

The linear system system $\mathbf{Ax}=\mathbf{c}$, where $\mathbf{A}$ is an $m\times n$ matrix, has:

• No solution if $\text{rank}(\mathbf{A})<\text{rank}(\mathbf{A}|\mathbf{c})$
• A unique solution if $\text{rank}(\mathbf{A})=\text{rank}(\mathbf{A}|\mathbf{c})$ and $\text{rank}(\mathbf{A})=n$
• Infinite solutions if $\text{rank}(\mathbf{A})=\text{rank}(\mathbf{A}|\mathbf{c})$ and $\text{rank}(\mathbf{A})<n$

Note that $\text{rank}(\mathbf{A})\le \text{rank}(\mathbf{A}|\mathbf{c})$