Question 14

Analyze the three statements for a system of linear equations $Ax = b$, where $A$ is an $m \times n$ matrix:

1. Statement I: "If $m < n$, then all such systems have a solution."
   An underdetermined system ($m < n$) can be inconsistent if the equations are contradictory. For example, in 3 variables with 2 equations:
   $$x_1 + x_2 + x_3 = 1$$
   $$x_1 + x_2 + x_3 = 2$$
   Here $m=2 < n=3$, but the system has no solution. Hence, Statement I is FALSE.

2. Statement II: "If $m > n$, then none of these systems has a solution."
   An overdetermined system ($m > n$) can be consistent if the equations are linearly dependent and compatible with $b$. For example, with $m=3, n=1$:
   $$x_1 = 2$$
   $$2x_1 = 4$$
   $$3x_1 = 6$$
   This has the unique solution $x_1 = 2$. Furthermore, any homogeneous system $Ax = 0$ always has the trivial solution $x = 0$. Hence, Statement II is FALSE.

3. Statement III: "If $m = n$, then there exists a system which has a solution."
   This is an existential statement. For any $m = n$, we can choose $A = I_n$ (the $n \times n$ identity matrix) and any vector $b$. The system $I_n x = b$ has the unique solution $x = b$. Therefore, such a system exists, and Statement III is TRUE.

Thus, only Statement III is true. This corresponds to option (C).