Let α and β be real numbers. Consider a 3 × 3 matrix A such that A2=3 A+α I. If A4=21 A+β I, then

Question

Let α and β be real numbers. Consider a 3 × 3 matrix A such that A2=3 A+α I. If A4=21 A+β I, then (A) β=-8 (B) β=8 (C) a=4 (D) α=1

Tardigrade Admin · Accepted Answer

A 2=3 A +α I A 3=3 A 2+α A A 3=3(3 A +α I )+α A A 3=9 A +α A +3 α I A 4=(9+α) A 2+3 α A =(9+α)(3 A +α I )+3 α A = A (27+6 α)+α(9+α) ⇒ 27+6 α=21 ⇒ α=-1 ⇒ β=α(9+α)=-8

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

$β = - 8$

$β = 8$

$a = 4$

$α = 1$

$A^{2} = 3 A + α I$
$A^{3} = 3 A^{2} + α A$
$A^{3} = 3 (3 A + α I) + α A$
$A^{3} = 9 A + α A + 3 α I$
$A^{4} = (9 + α) A^{2} + 3 α A$
$= (9 + α) (3 A + α I) + 3 α A$
$= A (27 + 6 α) + α (9 + α)$
$\Rightarrow 27 + 6 α = 21 \Rightarrow α = - 1$
$\Rightarrow β = α (9 + α) = - 8$

Q. Let α and β be real numbers. Consider a 3×3 matrix A such that A2=3A+αI. If A4=21A+βI, then

β=−8

β=8

a=4

α=1