GATE 2024 PH – Question 51
$A^\alpha$ and $B_\beta$ are contravariant and covariant vectors; repeated indices are summed. Which expressions are tensors?
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Correct answer: (A) $A^\alpha B_\beta$; (B) $A^\alpha B_\beta/(A^\gamma B_\gamma)$
Explanation
A **tensor** is an object whose components transform in a definite way under a change of coordinates.
- **A. $A^\alpha B_\beta$.** The outer product of a contravariant and a covariant vector is a rank-(1,1) tensor. ✓
- **B. $A^\alpha B_\beta/(A^\gamma B_\gamma)$.** The numerator is a mixed tensor and $A^\gamma B_\gamma$ is a **scalar** (an invariant contraction), so dividing by it (if nonzero) keeps the tensor transformation law. ✓
- **C. $A^\alpha/B_\beta$.** Component-wise division does not obey the tensor transformation rules. ✗
- **D. $A^\alpha+B_\beta$.** Adding a contravariant component to a covariant one (different index types) is not a tensor operation. ✗
Answer **A and B**.