Let p be a non-singular matrix, 1 + p + p2 + .... + pn = 0 (0 denotes the null matrix) then p

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Multiplication of Matrices

Question

Let p be a non-singular matrix, 1 + p + p² + .... + pⁿ = 0 (0 denotes the null matrix) then p^–1 =

– pⁿ

– (1 + p +...+ pⁿ)

none

pⁿ

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Given :

1 + p + p² ...... pⁿ = 0

applying sum of AP formula

$1. [\frac{p^{n + 1} - I}{p - I}] = 0$ {I = Identity matrix}

∴ pⁿ⁺¹ = I.

⇒ pⁿ. p = I

⇒ pⁿ . pp^–1 = I . p^–1 (post multiplication with p^–1)

⇒ p² = p^–1

⇒ A is correct.