JEE Advanced · 2010 · 11 Apr 2010 · Paper 1 · Matrices

For a linear system $A\vec{x}=\vec{b}$ to have exactly two distinct solutions is impossible.

A linear system can have:

• Exactly one solution (when A is invertible)
• No solution (when the system is inconsistent)
• Infinitely many solutions (when A is singular and the system is consistent)

A linear system cannot have exactly two (or any finite number greater than one) distinct solutions. This is because if $\overrightarrow{x_1}$ and $\overrightarrow{x_2}$ are two distinct solutions, then any linear combination $t\overrightarrow{x_1}+(1-t)\overrightarrow{x_2}$ for $t\in \mathbb{R}$ is also a solution, giving infinitely many solutions.

Therefore, the number of such matrices is 0.