What is the dimensionality of the null space of matrix A.
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What is the dimensionality of the null space of matrix A.
A = [3 2 -9; -6 -4 18; 12 8 -36]
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Answer: D. D) 2
The second row is -2 * Row 1, and the third row is 4 * Row 1. All rows are scalar multiples of the first, giving rank = 1. By the Rank-Nullity Theorem for a 3x3 matrix, nullity = 3 - 1 = 2.
Step-by-step Derivation:
Step 1: Write rows of A: R1 = [3, 2, -9], R2 = [-6, -4, 18] = -2R1, R3 = [12, 8, -36] = 4R1.
Step 2: Since R2 and R3 are linearly dependent on R1, the row space has dimension 1 (rank = 1).
Step 3: Number of columns n = 3.
Step 4: By the Rank-Nullity Theorem: Rank(A) + Nullity(A) = n => 1 + Nullity(A) = 3 => Nullity(A) = 2.