Let A be a nilpotent matrix of index m- If I - An - I - A - A2 - Am-1- where I is the identity matrix of order same as matrix A- then the value of n is

A is nilpotent matrix of index m. ⇒ A^m=O Let B = I + A + A^2 + ⋯+ A^m 1 Post multiplying with (I A) on both sides, B(I A) = (I + A + A^2 + ⋯+ A^m 1) (I A) ⇒ B( ...

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Standard XII

Mathematics

Derivative from First Principle

Let A be a ni...

Question

Let $A$ be a nilpotent matrix of index $m .$ If $(I - A)^{n} = I + A + A^{2} + \dots + A^{m - 1},$ where $I$ is the identity matrix of order same as matrix $A,$ then the value of $n$ is

m

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m - 1

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- 1

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1

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Solution

The correct option is C $- 1$
$A$ is nilpotent matrix of index $m .$
$\Rightarrow A^{m} = O$
Let $B = I + A + A^{2} + \dots + A^{m - 1}$
Post-multiplying with $(I - A)$ on both sides,
$B (I - A) = (I + A + A^{2} + \dots + A^{m - 1}) (I - A)$
$\Rightarrow B (I - A) = I + A + A^{2} + \dots + A^{m - 1} - (A + A^{2} + \dots + A^{m - 1} + A^{m})$
$\Rightarrow B (I - A) = I - A^{m}$
$\Rightarrow B (I - A) = I$
$\Rightarrow B = (I - A)^{- 1}$
$∴ n = - 1$

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Let A be a nilpotent matrix of index m. If (I−A)n=I+A+A2+⋯+Am−1, where I is the identity matrix of order same as matrix A, then the value of n is

The correct option is C −1A is nilpotent matrix of index m. ⇒Am=O Let B=I+A+A2+⋯+Am−1 Post-multiplying with (I−A) on both sides, B(I−A)=(I+A+A2+⋯+Am−1)(I−A) ⇒B(I−A)=I+A+A2+⋯+Am−1−(A+A2+⋯+Am−1+Am) ⇒B(I−A)=I−Am ⇒B(I−A)=I ⇒B=(I−A)−1 ∴n=−1

Let $A$ be a nilpotent matrix of index $m .$ If $(I - A)^{n} = I + A + A^{2} + \dots + A^{m - 1},$ where $I$ is the identity matrix of order same as matrix $A,$ then the value of $n$ is