GATE CS 2016 Set 2 Q04 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To determine which statements are correct, let us analyze each one with simple counterexamples or direct proofs:

• Statement I: "If $m < n$, then all such systems have a solution."
  This statement is false. A system with fewer equations than variables ($m < n$) can still be inconsistent.
  For example, let $m = 2$ and $n = 3$ (variables $x, y, z$): $$\begin{align*} x + y + z &= 1 \\ x + y + z &= 2 \end{align*}$$ Since these two equations contradict each other, the system has no solution.


• Statement II: "If $m > n$, then none of these systems has a solution."
  This statement is false. An overdetermined system ($m > n$) can have a unique solution or infinitely many solutions if the extra equations are consistent with the others.
  For example, let $m = 3$ and $n = 2$ (variables $x, y$): $$\begin{align*} x &= 1 \\ y &= 1 \\ x + y &= 2 \end{align*}$$ This system has the unique solution $(x, y) = (1, 1)$.


• Statement III: "If $m = n$, then there exists a system which has a solution."
  This statement is true. We only need to show the existence of at least one such system.
  For any $n$, the identity system: $$\begin{align*} x_1 &= 1 \\ x_2 &= 1 \\ &\vdots \\ x_n &= 1 \end{align*}$$ has $m = n$ equations and $n$ variables, and it clearly has the unique solution $(1, 1, \dots, 1)$.

Thus, only Statement III is true.