Discussion
Match the rank condition with the nature of the solution set.
1. Ranks equal and equal to the number of unknowns 2. Ranks equal but less than the number of unknowns 3. Augmented rank exceeds coefficient rank
P. No solution Q. Unique solution R. Infinitely many solutions
Answer: A
Correct answer: 1-Q, 2-R, 3-P
The three cases are exhaustive. Equal ranks mean the system is consistent, and then the common rank against the number of unknowns decides between one solution and infinitely many: equal gives uniqueness, less gives an \( (n-r) \)-parameter family. Unequal ranks - and the augmented rank can only be the larger - mean inconsistency and no solution at all.
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