let D= [((v 5)),(((1/3) m)) ] find number (v) and (m) such that D^2 =5I (I=identity matrix)


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Question Number 153227 by Eric002 last updated on 05/Sep/21

$l e t D = [\begin{matrix} v 5 \\ \frac{1}{3} m \end{matrix}] f i n d n u m b e r (v) a n d$ $(m) s u c h t h a t D^{2} = 5 I (I = i d e n t i t y m a t r i x)$
	Answered by puissant last updated on 05/Sep/21
	$D^2 = [((v^2 +(5/3) 5v+5m)),(((v/3)+(m/3) (5/3)+m^2 )) ]= [((5 0)),((0 5)) ] ⇒ { ((v^2 =5−(5/3))),((v+m=0)) :}, { ((v+m=0)),((m^2 =5−(5/3))) :} ⇒ m=−v v^2 =((10)/3) ⇒ v=(√((10)/3)) ; m=−(√((10)/3)).. or v=−(√(((10)/3) )) ; m=(√((10)/3))..$
	$D^{2} = [\begin{matrix} v^{2} + \frac{5}{3} 5 v + 5 m \\ \frac{v}{3} + \frac{m}{3} \frac{5}{3} + m^{2} \end{matrix}] = [\begin{matrix} 5 0 \\ 0 5 \end{matrix}]$ $\Rightarrow {\begin{cases} v^{2} = 5 - \frac{5}{3} \\ v + m = 0 \end{cases}, {\begin{cases} v + m = 0 \\ m^{2} = 5 - \frac{5}{3} \end{cases}$ $\Rightarrow m = - v$ $v^{2} = \frac{10}{3} \Rightarrow v = \sqrt{\frac{10}{3}}; m = - \sqrt{\frac{10}{3}} . .$ $o r v = - \sqrt{\frac{10}{3}}; m = \sqrt{\frac{10}{3}} . .$
	Commented by Eric002 last updated on 05/Sep/21

	$w e l l d o n e$
	Answered by King1 last updated on 06/Sep/21


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