Class 12 Determinants · Long Answer3 Markssymmetric_skew_decomposition
Class 12 Maths · Determinants · Long Answer #54
If A is symmetric, B is skew-symmetric and A+B=\(\begin{bmatrix}2 & 3\\5 & -1\end{bmatrix}\), find det(AB).
✅ Step-by-Step Solution
Final Answer: −18
- Transposing A+B gives A−B=[[2,5],[3,−1]].
- Add and subtract these equations to get A=[[2,4],[4,−1]] and B=[[0,−1],[1,0]].
- The determinants are det(A)=−2−16=−18 and det(B)=1. Thus det(AB)=det(A)det(B)=−18.
- Therefore det(AB)=−18.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Transposing A+B gives A−B=[[2,5],[3,−1]].
- [1 mark] Add and subtract these equations to get A=[[2,4],[4,−1]] and B=[[0,−1],[1,0]].
- [1 mark] The determinants are det(A)=−2−16=−18 and det(B)=1. Thus det(AB)=det(A)det(B)=−18.
- [0.5 marks] Therefore det(AB)=−18.
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