Question

If A^k = 0, for some value of k, (I – A)^p = I + A + A² + …. A^k–1, thus p is (A is nilpotent with index k).

• A. –1
• B. –2
• C. –3
• D. None of these

Answer 1

Answer: (A)

–1

Answer 2

Let B = I + A + A² + … + A^k–1

Post multiply both sides by (I – A), so that

B(I – A) = (I + A + A² + … + A^k–1) (I – A)

= I – A + A – A² + A² – A³ + … –A^k–1 + A^k–1 – A^k

= I – A^k = I, since A^k = 0

Ž B = (I – A)^–1

Hence (I – A)^–1 = I + A + A² + … + A^k–1.

Thus, p = –1.

Hence (1) is correct answer.