GATE IN 2024 | Question: 9

Question

A matrix $M$ is constructed by stacking three column vectors $v_{1}, v_{2}, v_{3}$ as

$$ M=\left[\begin{array}{lll} v_{1} & v_{2} & v_{3} \end{array}\right]$$

Choose the set of vectors from the following options such that $\operatorname{rank}(M)=3$.

• A. $v_{1}=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right], \quad v_{2}=\left[\begin{array}{r}0 \\ -1 \\ 0\end{array}\right], \quad v_{3}=\left[\begin{array}{r}1 \\ -1 \\ 1\end{array}\right]$
• B. $v_{1}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], \quad v_{2}=\left[\begin{array}{r}-1 \\ 0 \\ 1\end{array}\right], \quad v_{3}=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$
• C. $v_{1}=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right], \quad v_{2}=\left[\begin{array}{r}-1 \\ 0 \\ 1\end{array}\right], \quad v_{3}=\left[\begin{array}{r}1 \\ -1 \\ 1\end{array}\right]$
• D. $v_{1}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], \quad v_{2}=\left[\begin{array}{r}-1 \\ 1 \\ -1\end{array}\right], \quad v_{3}=\left[\begin{array}{r}0 \\ -1 \\ 0\end{array}\right]$

Answer 1

Option 1

Notice:

Column 1 = Column 3

So columns are dependent.

❌ Rank < 3

Option 2

Column 3 is the zero vector.

A zero vector is always dependent.

❌ Rank < 3

Option 3

Let's compute the determinant:

Since determinant ≠ 0,

✅ Rank = 3