The rank of the matrix [[0,0,-3],[9,3,5],[3,1,1]] is A. 0 B. 1 C. 2 D. 3

GATE 1994 · Engineering Mathematics · Rank of Matrix · medium

Answer: rank(M) = 2. Answer: C.

1. Set up and perform row swap: M = [[0,0,-3],[9,3,5],[3,1,1]]. Swap R1 and R3: [[3,1,1],[9,3,5],[0,0,-3]].
2. Eliminate R2 using R1: R2 -> [9,3,5] - 3*[3,1,1] = [9-9, 3-3, 5-3] = [0,0,2]. Matrix: [[3,1,1],[0,0,2],[0,0,-3]].
3. Eliminate R3 using R2: R3 -> [0,0,-3] + (3/2)*[0,0,2] = [0,0,-3+3] = [0,0,0]. Matrix: [[3,1,1],[0,0,2],[0,0,0]].
4. Count nonzero rows: Echelon form: [[3,1,1],[0,0,2],[0,0,0]]. Two nonzero rows => rank = 2.