Class 12 Matrices · Long Answer3 Marksmatrix_equality
Class 12 Maths · Matrices · Long Answer #63
For A=\(\begin{bmatrix}1 & -1\\2 & -1\end{bmatrix}\) and B=\(\begin{bmatrix}a & 1\\b & -1\end{bmatrix}\), find a,b if (A+B)²=A²+B².
✅ Step-by-Step Solution
Final Answer: a=1,b=4
- Expand (A+B)²=A²+AB+BA+B². The condition is therefore AB+BA=O.
- Compute AB=[[a−b,2],[2a−b,3]] and BA=[[a+2,−a−1],[b−2,1−b]]. Their sum is [[2a−b+2,1−a],[2a−2,4−b]].
- Equating entries to zero gives a=1 from 1−a=0 and b=4 from 4−b=0. The other two entries then also vanish.
- Thus a=1,b=4, and these values satisfy the full matrix equation.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Expand (A+B)²=A²+AB+BA+B². The condition is therefore AB+BA=O.
- [1 mark] Compute AB=[[a−b,2],[2a−b,3]] and BA=[[a+2,−a−1],[b−2,1−b]]. Their sum is [[2a−b+2,1−a],[2a−2,4−b]].
- [1 mark] Equating entries to zero gives a=1 from 1−a=0 and b=4 from 4−b=0. The other two entries then also vanish.
- [0.5 marks] Thus a=1,b=4, and these values satisfy the full matrix equation.
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