Class 12 Matrices · Long Answer3 Markssymmetric_and_skew_symmetric
Class 12 Maths · Matrices · Long Answer #102
Prove that all entries on the main diagonal of a real skew-symmetric matrix are zero.
✅ Step-by-Step Solution
Final Answer: All aᵢᵢ=0.
- Let A=[aᵢⱼ] be real and skew-symmetric, so Aᵀ=−A.
- For each index i, the (i,i) entry of Aᵀ is aᵢᵢ, while that of −A is −aᵢᵢ. Equality therefore gives aᵢᵢ=−aᵢᵢ.
- Adding aᵢᵢ gives 2aᵢᵢ=0. Division by 2 is valid over the real numbers.
- Thus aᵢᵢ=0 for every i, proving the diagonal is zero.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let A=[aᵢⱼ] be real and skew-symmetric, so Aᵀ=−A.
- [1 mark] For each index i, the (i,i) entry of Aᵀ is aᵢᵢ, while that of −A is −aᵢᵢ. Equality therefore gives aᵢᵢ=−aᵢᵢ.
- [1 mark] Adding aᵢᵢ gives 2aᵢᵢ=0. Division by 2 is valid over the real numbers.
- [0.5 marks] Thus aᵢᵢ=0 for every i, proving the diagonal is zero.
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