Let A be real matrix such that , where I and O are the identity and null matrices, respectively. If , where and are real constants, then is equal to : [2025]

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Solution

Q.
Let A be a $3 \times 3$ real matrix such that $A^{2} (A - 2 I) - 4 (A - I) = O$ , where $I$ and O are the identity and null matrices, respectively. If $A^{5} = α A^{2} + β A + γ I$ , where $α, β$ and $γ$ are real constants, then $α + β + γ$ is equal to : [2025]

1 20

2 76

3 12

4 4

Ans.
(3)

We have, $A^{2} (A - 2 I) - 4 (A - I) = O$

$\Rightarrow A^{3} - 2 A^{2} - 4 A + 4 I = O$

$\Rightarrow A^{3} = 2 A^{2} + 4 A - 4 I$ ... (i)

Now, $A^{4} = A \cdot A^{3} = A (2 A^{2} + 4 A - 4 I)$

$= 2 A^{3} + 4 A^{2} - 4 A = 2 (2 A^{2} + 4 A - 4 I) + 4 A^{2} - 4 A$ [From (i)]

$= 4 A^{2} + 8 A - 8 I + 4 A^{2} - 4 A = 8 A^{2} + 4 A - 8 I$

Similarly, $A^{5} = A \cdot A^{4} = A (8 A^{2} + 4 A - 8 I)$

$= 8 A^{3} + 4 A^{2} - 8 A = 8 (2 A^{2} + 4 A - 4 I) + 4 A^{2} - 8 A$ [From (i)]

$= 20 A^{2} + 24 A - 32 I$

$\Rightarrow α = 20, β = 24 and γ = - 32$

$∴ α + β + γ = 20 + 24 - 32 = 12$ .

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