GATE CS 1996 Q1.7 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To find the false statement, let us analyze each option individually:

• Option A (True): According to the Rouché–Capelli theorem, a system of linear equations $Ax = b$ is consistent (has at least one solution) if and only if the rank of the coefficient matrix $A$ is equal to the rank of the augmented matrix $[A \mid b]$.
• Option B (True): For a homogeneous system $Ax = 0$ where $m < n$ (fewer equations than variables), the rank of $A$ is at most $m$. The number of free variables (nullity) is given by: $$\text{Nullity} = n - \text{rank}(A) \ge n - m > 0$$ Since there is at least one free variable, the system must have infinitely many solutions.
• Option C (False): If $m = n$ and $b \neq 0$, the system $Ax = b$ has a unique solution only if $A$ is non-singular (i.e., $\text{rank}(A) = n$). If $A$ is singular ($\text{rank}(A) < n$), the system will either have infinitely many solutions or no solution at all.
  Counterexample: Let $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ and $b = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$. Here, $m = n = 2$ and $b \neq 0$, but the system has no solution because $\text{rank}(A) = 1$ while $\text{rank}([A \mid b]) = 2$.
• Option D (True): For a homogeneous system $Ax = 0$ with $m = n$ and $\text{rank}(A) = n$, the matrix $A$ is invertible. Multiplying both sides by $A^{-1}$ gives the unique trivial solution $x = 0$.

Thus, statement C is false.