Let p a non singular matrix 1 + p + p^2 + .... + p^n = O (O denot… - Vidyadip

MCQ

Let $p$ a non singular matrix $1 + p + {p^2} + .... + {p^n} = O$ ($O$ denotes the null matrix), then ${p^{ - 1}} = $

✓
${p^n}$
B
$ - {p^n}$
C
-$(1 + p + .... + {p^n})$
D
None of these

✓

Answer

Correct option: A.

${p^n}$

a
(a) $1 + p + {p^2} + .......... + {p^n} = O$
Pre-multiplying both sides by ${p^{ - 1}}$
==> ${p^{ - 1}} + I + Ip + ............ + {p^{n - 1}}I = O.{p^{ - 1}}$
==> ${p^{ - 1}} + I(1 + p + {p^2} + ......... + {p^{n - 1}}) = O$
==> ${p^{ - 1}} = - (1 + p + {p^2} + ......... + {p^{n - 1}})I$
==> ${p^{ - 1}} = {p^n}$

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