Discussion
Augmenting a 3 by 3 coefficient matrix of rank 2 raises the rank to 3. What follows for the system?
Answer: C
Correct answer: it has no solution
When the augmented rank exceeds the coefficient rank the elimination produces a row of the form \( 0 = c \) with \( c \neq 0 \), which no assignment of the unknowns can satisfy. The system is inconsistent. This is the one case in which the solution set is genuinely empty; every other rank combination gives either one solution or infinitely many.
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