Identifying Solution Types from RREF

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Identifying Solution Types from RREF Read off whether a system has a unique solution, infinitely many, or none — straight from its reduced row echelon form. An augmented matrix has already been reduced to reduced row echelon form (RREF). Classify its system as having a unique solution , infinitely many solutions , or no solution , then read off the answer: Nice work — you classified an RREF system end to end. The whole method is three quick reads off the matrix: Check consistency first: any row [\,0\ 0\ \ 0 c\,] with c 0 means 0 = c , a contradiction — the system has no solution , and nothing else matters. Find the pivots: a pivot is a leading 1; its column is a pivot column. Variable columns without a pivot are free variables . Read the verdict: consistent with a free variable → infinitely many ; consistent with a pivot in every variable column (no free variables) → unique . This example: consistent, with x_3 free, so it has infinitely many solutions : (x_1,x_2,x_3 ) = (3-2t,\ 5+t,\ t ) . The same three reads classify any linear system — and they power everything ahead: rank, column space, and the structure of solution sets.

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