If A ∈ M_(2×2) , det(A )≠ 0 , A^( 3) = A^2 +A ⇒ Find the values of det (2A −I )


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Question Number 202371 by mnjuly1970 last updated on 25/Dec/23

$I f A \in M_{2 \times 2}, d e t (A) \neq 0$ $, A^{3} = A^{2} + A \Rightarrow F i n d t h e$ $v a l u e s o f d e t (2 A - I)$
	Answered by witcher3 last updated on 25/Dec/23

	$\det (A) \neq 0 \Rightarrow A is inversible$ $\Rightarrow A^{-} . A^{3} = A^{-} (A^{2} + A)$ $\Rightarrow A^{2} = A + I$ $A^{2} - A = (A - \frac{I}{2}) (A - \frac{I}{2}) - \frac{I_{2}}{4}$ $\Rightarrow {(A - \frac{I}{2})}^{2} = \frac{5}{4} I_{2}$ $\Rightarrow \det^{2} (A - \frac{I}{2}) = {(\frac{5}{4})}^{2}$ $\Rightarrow \det (A - \frac{I}{2}) = \underline{+} \frac{5}{4}$ $\Rightarrow \det (2 (A - \frac{I}{2})) = \underline{+} 5$
	Commented by mnjuly1970 last updated on 25/Dec/23

	$p l e a s e r e c h e c k s i r . . .$ $d e t (2 A - I) = \pm 5 \underset{―}{\underset{⏟}{⋚}}$
	Commented by witcher3 last updated on 25/Dec/23

	$yest \det ({aI}_{n}) = a^{n} . thanx$
	Commented by mnjuly1970 last updated on 25/Dec/23

	$t h a n k s a l o t s i r . s o n i c e s o l u t i o n$
	Answered by Rajpurohith last updated on 26/Dec/23

	$S o A i s a n n i h i l a t e d b y t h e p o l y n o m i a l$ $p (x) = x^{3} - x^{2} - x = x (x^{2} - x - 1)$ $s o x = 0, α_{1}, α_{2}$ $w h e r e α_{1} a n d α_{2} a r e t h e r o o t s o f x^{2} - x - 1 = 0$ $s o d e t (A) = 0 a n d α_{1} + α_{2} = 1 a n d α_{1} α_{2} = - 1$ $t h e r o o t s o f a n n i h i l a t i n g p o l y n o m i a l o f$ $B = 2 A - I w i l l b e - 1, 2 α_{1} - 1, 2 α_{2} - 1$ $s o d e t (2 A - I) = - (2 α_{1} - 1) (2 α_{2} - 1)$ $= - [4 α_{1} α_{2} - 2 (α_{1} + α_{2}) + 1]$ $= - [- 4 - 2 (1) + 1] = 5$
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