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Business Mathematics and Statistics · Ch 1 — Applications of Matrices and Determinants (Rank, Cramer's Rule, Transition Probability Matrices)

Consistency of a System of Linear Equations — the Rank Method

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Consistency of a System of Linear Equations — the Rank Method

Setting up the test

For a system of nn linear equations in nn unknowns, written as AX=BAX=B, form the augmented matrix [A∣B][A|B] (the coefficient matrix with the constants column attached). Comparing ρ(A)\rho(A) (rank of the coefficient matrix alone) with ρ([A∣B])\rho([A|B]) (rank of the augmented matrix) reveals how many solutions the system has, without solving it first.

The three cases

ρ(A)=ρ([A∣B])=n  ⟹  a UNIQUE solution\rho(A)=\rho([A|B])=n \implies \text{a UNIQUE solution}

ρ(A)=ρ([A∣B])=r<n  ⟹  INFINITELY MANY solutions\rho(A)=\rho([A|B])=r<n \implies \text{INFINITELY MANY solutions}

ρ(A)≠ρ([A∣B])  ⟹  NO solution — the system is INCONSISTENT\rho(A)\neq\rho([A|B]) \implies \text{NO solution — the system is INCONSISTENT}

Note

Why the comparison works

ρ(A)≠ρ([A∣B])\rho(A)\neq\rho([A|B]) means the constants column adds a genuinely new, independent direction the coefficient equations alone cannot reach — exactly the case where the equations contradict each other. When the ranks match but fall short of nn, the equations agree with each other but leave at least one free parameter, so infinitely many combinations satisfy all of them.

Worked reasoning: telling the three cases apart …

Definition 1Consistent System

A system of linear equations with at least one solution, confirmed by $\rho(A) …

Definition 2Augmented Matrix

The coefficient matrix AA with the constants column BB attached, written [A∣B][A|B], used to test a system's c …

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