Question

$A$ is a $3\times 5$ real matrix of rank $2$. For the set of homogeneous equations $A x = 0$, where $0$ is a zero vector and $x$ is a vector of unknown variables, which of the following is/are true?

• A. The given set of equations will have a unique solution.
• B. The given set of equations will be satisfied by a zero vector of appropriate size.
• C. The given set of equations will have infinitely many solutions.
• D. The given set of equations will have many but a finite number of solutions.

Hint:

For any homogeneous linear system $Ax=0$, the number of free variables equals $\text{nullity}(A)=n-\text{rank}(A)$. A positive nullity always means infinitely many solutions.

Answer 1

The Correct Option is B, C

Step 1: Use the rank–nullity theorem.
The matrix $A$ is of size $3\times 5$, so the number of unknowns is $5$. The rank of $A$ is given as $2$. For any matrix, \[ \text{nullity}(A) = \text{number of variables} - \text{rank}(A). \] Thus, \[ \text{nullity}(A) = 5 - 2 = 3. \] This means the null space of $A$ is 3-dimensional. 
Step 2: Interpret solutions of the homogeneous system $Ax=0$. 
A homogeneous system always satisfies the zero vector solution: \[ x = 0 \quad \Rightarrow \quad Ax = 0. \] Thus option (B) is clearly correct. 
Since the nullity is $3>0$, the solution set of $Ax=0$ contains infinitely many vectors — a whole 3-dimensional subspace of $\mathbb{R}^5$. Therefore, the system does not have a unique solution. It has infinitely many solutions. Thus option (C) is correct. 
Step 3: Check the remaining options. 
(A) A unique solution occurs only when the nullity is 0. Here nullity is 3, so this is false. 
(D) A finite number of solutions cannot occur in a linear homogeneous system unless it's the trivial solution only. Here the solution set is infinite (a subspace), so (D) is false. 
Final Answer: (B), (C)