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Square Matrix

JEE Maths question with a full step-by-step solution.

Question
A square matrix PP satisfies P2=I−PP^2 = I - P, where II is the identity matrix. If Pn=5I−8PP^n = 5I - 8P, then nn is:
A44
B55
C66correct
D77
Solution
Step 1: Compute successive powers using P2=I−PP^2 = I - P
P3=P⋅P2=P(I−P)=P−P2=P−(I−P)=2P−IP^3 = P\cdot P^2 = P(I-P) = P - P^2 = P - (I-P) = 2P - I
P4=P⋅P3=2P2−P=2(I−P)−P=2I−3PP^4 = P\cdot P^3 = 2P^2 - P = 2(I-P) - P = 2I - 3P
P5=P⋅P4=2P−3P2=2P−3(I−P)=5P−3IP^5 = P\cdot P^4 = 2P - 3P^2 = 2P - 3(I-P) = 5P - 3I
P6=P⋅P5=5P2−3P=5(I−P)−3P=5I−8PP^6 = P\cdot P^5 = 5P^2 - 3P = 5(I-P) - 3P = 5I - 8P
 Step 2: Compare with the given expression Since P6=5I−8PP^6 = 5I - 8P, we obtain n=6n = 6. Answer: (3)
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