In our last lesson, we celebrated the power of Gaussian Elimination. We saw it as a perfect machine that takes a complex system and hands us a single, unique solution. That was a wonderful, clean example—a world where every problem has one perfect answer.
Today, we step into the real world.
The true genius of Gaussian Elimination is not just that it finds answers, but that it is a powerful diagnostic tool. It tells us the nature of our system. It reveals whether our problem has that one perfect answer, or if it's a problem with no answer at all, or a problem with an entire family of answers.
There are only three possibilities for any system Ax=b. Let’s become detectives and learn to identify the clues for each one.
The Clues: Pivots and Contradictions
The entire story is told by the pivots—the first non-zero entry in each row after you've reached row echelon form. Everything hinges on two questions:
Do we encounter a mathematical contradiction during elimination?
After elimination, does every variable's column have a pivot, or are some "free"?
Case 1: No Solution — "The Contradiction"
This is the case where the equations are fundamentally at odds with each other. They describe a geometric situation that is impossible.
10011021031−1
The Diagnostic Rule: A Contradiction
The last row translates to 0x+0y+0z=−1, or 0=−1. This is a mathematical impossibility. If you find a row of `[0 0 ... | non-zero]`, the system has NO SOLUTION.
Geometric Meaning: The three planes defined by the equations never meet at a single point. In the column picture, the target vector b is not in the span of the columns of `A`.
Case 2: Infinite Solutions — "The Free Variable"
This happens when there are no contradictions, but some equations are redundant, giving us freedom.
100−2101104−10
The last row, `0=0`, is true but useless. The column for `z` does not have a pivot. This makes `z` a free variable.
The Diagnostic Rule: Free Variables
If there are no contradictions and at least one column has no pivot, the variable for that column is "free". You can choose its value, leading to INFINITELY MANY SOLUTIONS.
The General Solution is a Line:
xyz=2−10+t−3−11
Geometric Meaning: The three planes intersect along a single line. In the column picture, the columns of `A` are linearly dependent, and there are infinite ways to combine them to create `b`.
Case 3: A Unique Solution
This is the clean case we saw in the last lesson.
2001−801−215−122
The Diagnostic Rule: Full Pivots
If there are no contradictions and every variable's column has a pivot, the system has a UNIQUE SOLUTION. There are no free variables, so there is no freedom.
Geometric Meaning: The planes intersect at a single point. In the column picture, the columns of `A` are linearly independent and `b` is in their span.