Let and be real numbers. Consider a 3 × 3 matrix A such that A 2 = 3 A + I . If A 4 = 21 A + I , then [2023]

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Solution

Q.
Let $α$ and $β$ be real numbers. Consider a $3 \times 3$ matrix A such that $A^{2} = 3 A + α I .$ If $A^{4} = 21 A + β I,$ then [2023]

1 $β = - 8$

2 $β = 8$

3 $α = 4$

4 $α = 1$

Ans.
(1)

$Given, A^{2} = 3 A + α I and A^{4} = 21 A + β I$

$Now, A^{2} A^{2} = (3 A + α I) (3 A + α I)$

$= 9 A^{2} + 6 α I A + α^{2} I = 9 (3 A + α I) + 6 α I A + α^{2} I$

$= 27 A + 9 α I + 6 α A + α^{2} I = (27 + 6 α) A + (9 α + α^{2}) I$

$So, 27 + 6 α = 21 \Rightarrow α = - 1$

$\Rightarrow 9 α + α^{2} = β \Rightarrow 9 (- 1) + {(- 1)}^{2} = β \Rightarrow β = - 8$

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