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Question: Let <img src="https://cdn.pureessence.tech/canvas_173.png?top_left_x=423&top_left_y=300&width=273&he...

Let be a non-singular matrix,

1+p+p2+.......+pn=01 + p + p^{2} + ....... + p^{n} = 0 (0 denotes the null matrix), then p−1p^{- 1}=

A

pnp^{n}

B

−pn- p^{n}

C

−(1+p+.....+pn)- (1 + p + ..... + p^{n})

D

None

Answer

pnp^{n}

Explanation

Solution

We have, 1+p+p2+−−−+pn=01 + p + p^{2} + - - - + p^{n} = 0

Multiplying both sides by p−1p^{- 1}, p−1+I+Ip+.....+pn−1I=0.p−1p^{- 1} + I + Ip + ..... + p^{n - 1}I = 0.p^{- 1}

p−1+I(1+p+....+pn−1)=0p^{- 1} + I(1 + p + .... + p^{n - 1}) = 0 ⇒ p−1=−I(1+p+p2+....+pn−1)=−(−pn)=pnp^{- 1} = - I(1 + p + p^{2} + .... + p^{n - 1}) = - ( - p^{n}) = p^{n}.