UPSEE Solved Model Paper 1

(𝐴 − 2𝐼)(𝐴 − 𝛼𝐼) = 0
⇒ 𝐴 ⋅ 𝐴 − (2 + 𝛼)𝐴 ⋅ 𝐼 + 2𝛼𝐼 ⋅ 𝐼 = 0
⇒ 𝐴2 − (2 + 𝛼)𝐴𝐼 + 2𝛼𝐼 = 0
𝑃𝑟𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 𝐴−1 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒:
⇒ 𝐴−1 ⋅ 𝐴 ⋅ 𝐴 − (2 + 𝛼) 𝐴−1 ⋅ 𝐴 ⋅ 𝐼 + 2𝛼 𝐴−1 ⋅ 𝐼 = 0
⇒ 1. 𝐴 − (2 + 𝛼)𝐼 + 2𝛼 𝐴−1 𝐼 = 0
⇒ 𝐴 − (2 + 𝛼)𝐼 + 2𝛼 𝐴−1 𝐼 = 0
⇒ 𝐴 + 2𝛼 𝐴−1 = (2 + 𝛼)𝐼 ⋯ (𝑖)
Now,
𝐴 + 𝐴−1 =

5

2

I
Compare it with equation (i)
2𝛼 = 1
α =

1

2