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

The correct option is A
$n A - (n - 1) I$

$A^{2} = 2 A - I A^{3} = A^{2} . A = (2 A - I) A = 2 A^{2} - I A = 2 (2 A - I) - A A^{3} = 3 A - 2 I M u l t i p l y i n g w i t h A, A^{4} = A^{3} . A = (3 A - 2 I) A = 3 A^{2} - 2 A = 3 (2 A - I) - 2 A A^{4} = 4 A - 3 I I n g e n e r a l, A^{n} = n A - (n - 1) I$

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Similar questions

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

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