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Definition of a Determinant

A square matr...

Question

A square matrix P satisfies $P^{2} = I - P,$ where I is an identity matrix of order as order of P. if $P^{n} = 5 I - 8 P, t h e n n =$

A

4

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B

5

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C

6

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D

7

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Solution

The correct option is C 6
Since $P^{2} = I - P (G i v e n) - (1)$
$P^{3} = P (I - P)$
$P^{3} = P - P^{2} = P - (I - P)$
$P^{3} = 2 P - I$
Similarly $P^{4} = 2 P^{2} - P = 2 I - 3 P$ AND $P^{5} = 5 P - 3 I$
$P^{6} = 5 P^{2} - 3 P = 5 I - 8 P$
$∴ n = 6$

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P

be a square matrix satisfying

P^{2} = I - P,

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is identity matrix. If

P^{n} = 5 I - 8 P,

then the value of

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be a square matrix satisfying

P^{2} = I - P,

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P^{n} = 5 I - 8 P,

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be an identity matrix of order

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P^{n} = 5 I - 8 P

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