Consistency of System of Linear Equations

1. The System in Matrix Form

A system of m linear equations in n unknowns is written AX = B, where A is the m × n coefficient matrix, X the n × 1 column of unknowns and B the m × 1 column of constants. The augmented matrix is [A | B], obtained by attaching the constants as an extra column.

AX = B ⇔ a_1 x_1 + a_2 x_2 + ... + a_n x_n = b

Homogeneous system: B = 0 (all constants zero).
Non-homogeneous system: B ≠ 0.
A system is consistent if it has at least one solution, inconsistent if it has none.

2. The Consistency Test (Rouché–Capelli Theorem)

Compare the rank of the coefficient matrix with the rank of the augmented matrix:

Condition	Conclusion
ρ(A) ≠ ρ(A \| B)	Inconsistent — no solution (the extra equation contradicts the others)
ρ(A) = ρ(A \| B) = n (n = number of unknowns)	Consistent with a unique solution
ρ(A) = ρ(A \| B) = r < n	Consistent with infinitely many solutions, depending on (n − r) free parameters

Inconsistency always shows up during elimination as a row of the form [0 0 … 0 | c] with c ≠ 0 — i.e. the impossible statement 0 = c.

3. Homogeneous Systems — Always Consistent

For AX = 0, the zero vector X = 0 is always a solution (the trivial solution), so a homogeneous system is never inconsistent. What matters is whether anything else exists:

ρ(A) = n ⇒ only the trivial solution X = 0 (unique solution, the zero solution);
ρ(A) = r < n ⇒ non-trivial (infinitely many) solutions, with (n − r) free parameters.
So a homogeneous system has non-trivial solutions if and only if det A = 0 (for a square A) — a result used constantly in eigenvalue problems in the coming topics.

  x +  y +  z =  6
  x + 2y + 3z = 14
  x + 4y + 9z = 36

        | 1  1  1 |
   det  | 1  2  3 | = (2-1)(3-1)(3-2) = 2 ≠ 0   (Vandermonde-type determinant)
        | 1  4  9 |
   ⇒  ρ(A) = ρ(A|B) = 3 = n  ⇒  UNIQUE SOLUTION

   Elimination:  R2 → R2 - R1 , R3 → R3 - R1 :
        [ 1  1  1 |  6 ]
        [ 0  1  2 |  8 ]
        [ 0  3  8 | 30 ]      R3 → R3 - 3R2 ⇒  [ 0  0  2 |  6 ]

   Back substitution:  2z = 6 ⇒ z = 3 ;  y + 2z = 8 ⇒ y = 2 ;  x + y + z = 6 ⇒ x = 1
   Check: 1 + 2 + 3 = 6 ✓   1 + 4 + 9 = 14 ✓   1 + 8 + 27 = 36 ✓

8. Methods Recap

Situation	Best method
Square system, det ≠ 0 needed	Determinant (Cramer's rule or x = A⁻¹B)
Any system, want to know existence first	Rank test on [A \| B]
Need the actual solutions	Gauss–Jordan elimination + back substitution
Homogeneous system	Rank test: non-trivial solutions ⇔ ρ(A) < n ⇔ det A = 0 (square case)

9. Exam Points

State the condition for consistency of AX = B in terms of ranks of A and [A | B] — 5 marks, almost certain.
Discuss the consistency of a given 3 × 3 system and solve it if consistent — 5-7 marks, very frequent.
Show that a homogeneous system always has the trivial solution and find non-trivial solutions when they exist — 5 marks, frequent.
Find the values of k for which the system has a unique / infinite / no solution — 7 marks, very frequent.
Geometric interpretation of the three cases for 2 and 3 unknowns (intersecting lines, coincident lines, parallel lines; planes) — 3-5 marks.
When does a square homogeneous system possess a non-trivial solution? — 3 marks.

Consistency of System of Linear Equations