99 Percentile Qs Bank for JEE MainMathematicsMatrices
If P = 3 2 1 2 - 1 2 3 2 ,   A = 1 1 0 1 and Q = P A P ⊤ and X = P ⊤ Q 2005 P , then X is equal to
Options
- A1 2005 0 1
- B4 + 2005 3 6015 2005 4 - 2005 3
- C1 4 2 + 3 1 - 1 2 - 3
- D1 4 2005 2 - 3 2 + 3 2005
Correct answer
A. 1 2005 0 1
Step-by-step solution
Given P = 3 2 1 2 - 1 2 3 2 ,   A = 1 1 0 1 and Q = P A P ⊤ and X = P ⊤ Q 2005 P Now, P P T = 3 2 1 2 - 1 2 3 2 3 2 - 1 2 1 2 3 2 = 3 2 × 3 2 + 1 2 × 1 2 3 2 × - 1 2 + 1 2 × 3 2 - 1 2 × 3 2 + 1 2 × 3 2 - 1 2 × - 1 2 + 3 2 × 3 2 = 1 0 0 1 = I ⇒ P P T = I   . . . . . . . . . . . . . . i Also, Q = P A P ⊤ ⇒ Q 2 = P A P ⊤ P A P ⊤ ⇒ Q 2 = P A P ⊤ P A P ⊤ ⇒ Q 2 = P A I A P ⊤ (from equation i ) ⇒