If A 2-3 A -2 I and A 8- aA - bI- then a - b isA- 7B- -1C- 0D- 1

The correct option is D 1We have, A^2 = 3A 2I ⇒ A^3 = 3A^2 2A =3(3A 2I) 2A ⇒ A^3 =7A 6I ⇒ A^4 =7A^2 6A ⇒ A^4 =7(3A 2I) 6A ⇒ A^4 =15A 14I So A^n =(2^n ...

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Standard XII

Mathematics

Addition and Subtraction of a Matrix

If A 2=3 A -2...

Question

If $A^{2} = 3 A - 2 I$ and $A^{8} = a A + b I$ , then $a + b$ is

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Solution

The correct option is C 1
We have, $A^{2} = 3 A - 2 I \Rightarrow A^{3} = 3 A^{2} - 2 A = 3 (3 A - 2 I) - 2 A$
$\Rightarrow A^{3} = 7 A - 6 I$
$\Rightarrow A^{4} = 7 A^{2} - 6 A$
$\Rightarrow A^{4} = 7 (3 A - 2 I) - 6 A$
$\Rightarrow A^{4} = 15 A - 14 I$

So $A^{n} = (2^{n} - 1) A - (2^{n} - 2) I$

So $A^{8} = 255 A - 254 I$

$∴ a + b = 1$

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If A2=3A−2I and A8=aA+bI, then a+b is

The correct option is C 1We have, A2=3A−2I⇒A3=3A2−2A=3(3A−2I)−2A ⇒A3=7A−6I ⇒A4=7A2−6A ⇒A4=7(3A−2I)−6A ⇒A4=15A−14I So An=(2n−1)A−(2n−2)I So A8=255A−254I ∴a+b=1

If $A^{2} = 3 A - 2 I$ and $A^{8} = a A + b I$ , then $a + b$ is