GATE 2022 IN Q.24: solution and answer

Official answer: A, DSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

1. Before expanding anything, look for proportional rows. Row 3 is $(4,6,14)=2\times (2,3,7)$, exactly twice Row 1: $R_3=2R_1$.

2. Apply the row operation $R_3\to R_3-2R_1$. Adding a multiple of one row to another changes neither the rank nor the determinant:
   $$M\sim \left[\begin{array}{ccc}2& 3& 7\\ 6& 4& 7\\ 0& 0& 0\end{array}\right]$$

3. A square matrix with a zero row has determinant 0, so $\det M=0$ and (D) is correct. Since $\det M=0$, the rank of this $3\times 3$ matrix is less than 3, so (B) is false.

4. $R_3-2R_1=0$ is a non-trivial combination of the rows that equals the zero vector, so the rows are linearly dependent and (C) is false.

5. Rows 1 and 2, $(2,3,7)$ and $(6,4,7)$, are not multiples of each other. The $2\times 2$ minor $\left|\begin{array}{cc}2& 3\\ 6& 4\end{array}\right|=8-18=-10\ne 0$ confirms this, so the rank is at least 2.

6. Combining the two bounds, $\rho (M)=2$, so (A) is correct. The correct options are (A) and (D).