Q.32 What is the rank of the following matrix?
| 5 | 3 | −1 |
| 6 | 2 | −4 |
| 14 | 10 | 0 |
The rank of a matrix is one of the most important concepts in
linear algebra. It helps determine the number of linearly independent
rows or columns and is widely tested in exams such as GATE,
CSIR-NET, and IIT-JAM.
Given Matrix
A =
| 5 3 -1 |
| 6 2 -4 |
| 14 10 0 |
Concept Used
- The rank of a matrix is the maximum number of linearly independent rows or columns.
- For a square matrix, if the determinant is non-zero, the rank equals the order of the matrix.
- If the determinant is zero, the rank is less than the order.
Step-by-Step Solution
Step 1: Find the determinant
|A| =
= 5(2×0 − (−4×10))
− 3(6×0 − (−4×14))
− 1(6×10 − 2×14)
Step 2: Simplify
= 5(40) − 3(56) − (60 − 28)
= 200 − 168 − 32
= 0
Step 3: Interpret determinant
Since the determinant is zero, the rank of the matrix is less than 3.
Step 4: Check linear dependence
Row3 = 2(Row1 + Row2)
Hence, only two rows are linearly independent.
Correct Answer
Option (C): 2
Key Concept Summary
- Determinant ≠ 0 ⇒ Full rank
- Determinant = 0 ⇒ Rank less than order
- Rank equals number of linearly independent rows or columns
Conclusion
The determinant of the given matrix is zero, but two rows are linearly independent.
Therefore, the rank of the matrix is 2, making
Option (C) the correct answer.
