Consistency of Systems of Linear Equations
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Direct answer
A linear system's fate is decided by comparing two ranks: that of the coefficient matrix A and that of the augmented matrix [A | B]. If rank(A) = rank([A | B]) = n (the number of unknowns), the system has exactly one solution; if the ranks are equal but fall short of n, there are infinitely many solutions, with n - r free parameters; if rank(A) < rank([A | B]), no solution exists. Homogeneous systems AX = 0 are always consistent — the trivial solution — and gain nontrivial solutions exactly when rank(A) < n, which for a square system reads det A = 0. For a 2 × 2 or invertible case, Cramer's rule delivers the unique solution as ratios of determinants: x = D1/D, y = D2/D with D = det A ≠ 0.
What you must remember
- The three-case rule: rank(A) = rank([A|B]) = n gives a unique solution; equal ranks r < n give infinitely many (n - r parameters); rank(A) < rank([A|B]) gives no solution.
- Homogeneous always consistent: AX = 0 has the trivial solution; nontrivial solutions exist iff rank(A) < n, equivalent to det A = 0 for square systems.
- Cramer's rule: for det D ≠ 0, each unknown equals the determinant of the matrix with that column replaced by B, divided by D.
- Geometric reading (3 unknowns): each equation is a plane; unique solution = three planes at one point, infinite solutions = planes sharing a line or coinciding, no solution = parallel planes or a triangular prism arrangement.
- Two-equation cross-ratio test: a1x + b1y = c1 and a2x + b2y = c2 have a unique solution when a1/a2 ≠ b1/b2, infinitely many when a1/a2 = b1/b2 = c1/c2, none when a1/a2 = b1/b2 ≠ c1/c2.
- Parameter counting: with rank r and n unknowns, the solution family carries n - r free constants — rank 2 with 3 unknowns means a one-parameter (line) family.
- Row-reduction protocol: echelon-reduce [A | B] and read both ranks from the same matrix; the last-column entries decide whether the augmented rank climbs above the coefficient rank.
Two systems that differ by one constant
Take the system x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30. Row-reduce the augmented matrix: R2 - R1 gives (0, 1, 2 | 8); R3 - R1 gives (0, 3, 6 | 24); then R3 - 3R2 gives (0, 0, 0 | 0). The augmented matrix has two nonzero rows, and so does the coefficient part: rank(A) = rank([A|B]) = 2 < 3 unknowns — infinitely many solutions with one parameter. Indeed the second equation minus the first reads y + 2z = 8, and the third is three times that, a redundancy that leaves a free variable. Now change the last constant from 30 to 32: the final row operation yields (0, 0, 0 | 2) — a row asserting 0 = 2. The coefficient rank stays 2 while the augmented rank climbs to 3, so the system is inconsistent. One constant moved, and the geometry shifted from three planes through a common line to a triangular prism (each pair meets, all three never do). This is the entire theory in one example, and writing the (0, 0, 0 | c) row explicitly is how full marks are secured.
How the exam frames it
JEE Main asks for the value of a parameter (k or λ) that makes a given system consistent, or the number of solutions when ranks are given — the determinant route (set det A = 0, then verify which value survives) is the expected two-step: for det = 0 candidates, substitute back and check the augmented rank. Advanced phrases the same theory through planes (for what value of k do the planes meet in a point?), asks for the solution set of a rank-deficient system in parametric form, or connects to eigenvalue questions where (A - λI)X = 0 gains nontrivial solutions. The recurring errors: concluding consistency from det A = 0 alone (it must be the augmented matrix that drops rank — det A = 0 is necessary but not sufficient for infinitely many); counting unknowns from the number of equations when they differ; and in Cramer questions, replacing the wrong column for the numerator determinants. Homogeneous systems with more unknowns than equations are always rank-deficient — a one-line argument (rank ≤ m < n) that Main reuses in matrix-theory questions. Systems of equations sit in the matrices-and-determinants unit of both syllabi.
Frequently asked questions
What is the condition for a unique solution to AX = B?
rank(A) = rank([A | B]) = n, the number of unknowns; for a square system this is the same as det A ≠ 0.
When is a system inconsistent?
When rank(A) < rank([A | B]) — the augmented matrix has a row of the form (0, 0, ..., 0 | c) with c ≠ 0, asserting an impossibility.
Why is a homogeneous system always consistent?
Because X = 0 satisfies it identically; the only question is whether nontrivial solutions exist, which happens exactly when rank(A) < n.
How many free parameters does a consistent system with rank r have?
n - r, one for each unknown beyond the rank — rank 2 in three unknowns describes a one-parameter family of solutions along a line.
How does Cramer's rule solve a 2 × 2 system?
Compute D = det A; then x = D1/D and y = D2/D, where D1 and D2 replace the respective columns of A by the constants — valid only while D ≠ 0.