CUET UG 2025 — Mathematics Algebra

A matrix $A$A is skew-symmetric if:

$$A^T = -A$$AT=−A

For any element $a_{ij}$aij​ in the matrix:

$$a_{ij} = -a_{ji}$$aij​=−aji​

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For diagonal elements where $i = j$i=j:

$$a_{ii} = -a_{ii}$$aii​=−aii​

$$a_{ii} + a_{ii} = 0$$aii​+aii​=0

$$2a_{ii} = 0$$2aii​=0

$$a_{ii} = 0$$aii​=0

All diagonal elements must be zero.

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Option 1: $a_{ij} = \frac{1}{a_{ji}}$aij​=aji​1​

This is not a property of skew-symmetric matrices.

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Option 2: $a_{ij} = 0$aij​=0 for $i = j$i=j

This is correct. All diagonal elements (where $i = j$i=j) must be zero.

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Option 3: $a_{ij} = 0$aij​=0 for all $i, j$i,j

This is incorrect. Only diagonal elements are zero. Off-diagonal elements can be non-zero.

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Option 4: $a_{ij} \neq 0$aij​=0 for $i = j$i=j

This is incorrect. Diagonal elements must be zero.

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Therefore, the answer is Option 2.