A square matrix P satisfies P^2 = I, -, P ; If P^n = 5I, -, 8P then n is

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3 and 4 .Determinants and Matrices

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A square matrix $P$ satisfies $P^2 = I\, -\, P$ ; If $P^n = 5I\, -\, 8P$ then $n$ is

A

$4$

B

$5$

C

$6$

D

$7$

Solution

$P^{4}=(I-P)(I-P)=I-2 P+P^{2}=2 I-3 P$

${{\text{P}}^6} = (2{\text{I}} – 3{\text{P}})({\text{I}} – {\text{P}}) = 5{\text{I}} – 8{\text{P}} \Rightarrow \boxed{n = 6}$

Standard 12

Mathematics

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