If $A$ is a $3 \times 3$ matrix with rank 2, the dimension of $\ker A$ is
A0
B1
C2
D3
Answer & Solution
Correct answer: B. 1
1. RANK-NULLITY: $\dim(\text{domain}) = \text{rank} + \text{nullity}$.
2. For $A : \mathbb{R}^3 \to \mathbb{R}^3$ with rank 2:
$3 = 2 + \text{nullity}$
3. So nullity = $\dim(\ker A) = 1$.
4. Geometrically: the matrix sends $\mathbb{R}^3$ to a 2-dim plane (rank 2), and the 1-dim line of vectors mapped to 0 is the kernel.
5. Other options violate rank-nullity.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §3.3 (Rank-nullity application)._
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