Question related to system of homogeneous system of equations.

Question

Consider the system of equation MX=0 , where M is a n by n matrix. Then

A. System may not have solution.

B. System may have exactly one solution.

C. System may have exactly two solutions.

D. System may have infinitely many solutions.

No information other than the n by n matrix is given. I know that 1st option is not true because homogeneous system always has trivial solution. Option 2nd is false because if homogeneous equation has solution more than one than its definitely infinitely many that cannot be exactly two solutions. I am confused between B and D options.

Answer 1

The answer seems to be both B and D. It is possible for the only solution to be $0$ (i.e. when your matrix is of full rank and hence invertible). Another possibility is that if your matrix is not of full rank there will be a solution $ \mathbf {v} \neq 0$ to your equation, i.e. $M \mathbf{v} = 0$. Then for any $\lambda \in \mathbb{R}$ we have $M (\lambda \mathbf {v} )= \lambda M \mathbf{v} = \lambda 0 = 0$. So in fact any vector of the form $ \lambda \mathbf{v}$ is a solution and since $\mathbf{v} \neq 0$ there will be infinitely many.