A matrix A is such that A 2-2 A - I- where I is unity matrix- then for n - 2- A n is equal toA- n A-n-1 IB- n A-I - 2 n -1 A - n -1 ID- 2 n A - nI

The correct option is A nA (n 1)IA^2=2A I Let us calculate A^3 nbsp; A^3=A^2A ⇒ A^3=(2A I)A nbsp; nbsp; ⇒ A^3=2A^2 A ⇒ nbsp;A^3=2(2A I) A ⇒ A^3=3A 2I ...

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Standard XII

Mathematics

Transpose of a Matrix

A matrix A is...

Question

A matrix $A$ is such that $A^{2} = 2 A - I$ , where $I$ is unity matrix, then for $n \geq 2$ , $A^{n}$ is equal to

n A - (n - 1) I

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n A - I

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2^{n - 1} A - (n - 1) I

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2^{n} A - n I

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Solution

The correct option is A $n A - (n - 1) I$
$A^{2} = 2 A - I$
Let us calculate $A^{3}$
$A^{3} = A^{2} A$
$\Rightarrow A^{3} = (2 A - I) A \Rightarrow A^{3} = 2 A^{2} - A \Rightarrow A^{3} = 2 (2 A - I) - A \Rightarrow A^{3} = 3 A - 2 I$
Lets calculate $A^{4}$
$A^{4} = 3 A^{2} - 2 A \Rightarrow A^{4} = 3 (2 A - I) - 2 A \Rightarrow A^{4} = 4 A - 3 I$
Similarly we can write,
$A^{n} = n A - (n - 1) I$

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A matrix A is such that A2=2A−I, where I is unity matrix, then for n≥2, An is equal to

The correct option is A nA−(n−1)IA2=2A−I Let us calculate A3 A3=A2A ⇒A3=(2A−I)A⇒A3=2A2−A⇒A3=2(2A−I)−A⇒A3=3A−2I Lets calculate A4 A4=3A2−2A⇒A4=3(2A−I)−2A⇒A4=4A−3I Similarly we can write, An=nA−(n−1)I

A matrix $A$ is such that $A^{2} = 2 A - I$ , where $I$ is unity matrix, then for $n \geq 2$ , $A^{n}$ is equal to