Matrix A is such that A 2-2 A - I- where I is the Identity matrix- Then for n - 2- A n is equal toA- n A-IB- 2n-1 A-n-1 IC- 2n-1 A-ID- nA - n -1 I

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Equality of Matrices

Matrix A is s...

Question

Matrix A is such that $A^{2} = 2 A - I,$ where I is the Identity 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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$2^{n - 1} A - I$

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Solution

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A

A^{2} = 2 A - I

I

n \geq 2

A^{n}

A^{2}

(I + A)^{n}

If A = (\begin{matrix} 1 & 0 1 & 1 \end{matrix}] and I = (\begin{matrix} 1 & 0 0 & 1 \end{matrix}], then which of the following holds for all n \in N ?

A

A^{2} = 2 A - I

I

n \geq 2

n \in N; A^{n}

X^{2} = 2 X - I

n \geq 2, X^{n}

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