A homogeneous system of linear equations $Ax = 0$ has $4$ equations and $6$ unknowns. If the rank of the coefficient matrix $A$ is $3$, the number of linearly independent solutions is:
Options:
(A) 1
(B) 2
(C) 3
(D) 4
Answer Key:
C
Step-by-Step Solution & Explanation
By the Rank-Nullity Theorem:
$\operatorname{rank}(A) + \operatorname{nullity}(A) = n$
where $n$ is the number of unknowns (columns of $A$).
Here $n = 6$ and $\operatorname{rank}(A) = 3$:
$\operatorname{nullity}(A) = n - \operatorname{rank}(A) = 6 - 3 = 3$
The nullity of $A$ represents the dimension of the solution space, which is the number of linearly independent solutions. Therefore, the answer is $3$.