GATE 2021 CH – Question 30
$A,B,C$ and $D$ are vectors of length 4: $A=[a_1\ a_2\ a_3\ a_4]$, $B=[b_1\ b_2\ b_3\ b_4]$, $C=[c_1\ c_2\ c_3\ c_4]$, $D=[d_1\ d_2\ d_3\ d_4]$. It is known that B is not a scalar multiple of A. Also, C is linearly independent of A and B. Further, $D=3A+2B+C$. The rank of the matrix with rows A, B, C and D is _____.
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Correct answer: 3
Explanation
A and B are independent as intended by the stated non-multiple condition, and C adds a third independent row. The fourth row is explicitly $D=3A+2B+C$, so it adds no new direction.
The row operation $R_4\leftarrow R_4-3R_1-2R_2-R_3$ makes the fourth row zero, leaving three independent rows. **Rank = 3.**