GATE Civil 2021 Set 2 | Question: 2

Question

The rank of the matrix $\begin{bmatrix} 5 & 0 & -5 & 0\\ 0 & 2 & 0 & 1\\ -5 & 0 & 5 & 0\\ 0 & 1 & 0 & 2 \end{bmatrix}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Step-by-Step Calculation:

1. Analyze Row 1 and Row 3: Notice that $R_3$ is just $R_1$ multiplied by $-1$.

   • Operation: $R_3 \to R_3 + R_1$

   • Result: The third row becomes $[0, 0, 0, 0]$.

2. Analyze Row 2 and Row 4: These rows are $[0, 2, 0, 1]$ and $[0, 1, 0, 2]$.

   • These are linearly independent because one is not a scalar multiple of the other (the ratios $2/1$ and $1/2$ are not equal).

3. Count Independent Rows: * Row 1 is independent.

   • Row 2 is independent.

   • Row 3 is now all zeros (dependent).

   • Row 4 is independent.

4. Final Count: There are 3 linearly independent rows.

Final Answer: Option C (3)