Question

Dadas as matrizes $A\left[\begin{array}{ccc}3&x\\x&x^{2} \\\end{array}\right]$ e $B\left[\begin{array}{ccc}6x&x\\2x&1\\\end{array}\right]$, com x ∈ R, quais os valores de x que tornam verdadeira a igualdade detA= 3 detB ?

Answer 1

Os valores de x que tornam verdadeira a igualdade $det~A=3~.~det~B$ são $\bold{x=0}$  e  $\bold{x=\frac{9}{4}=2,25}$

• Resolvendo o problema

$det~A=3~.~x^2-x~.~x\\\\det~A=3x^2-x^2\\\\det~A=2x^2\\\\\\det~B=6x~.~1-x~.~2x\\\\det~B=6x-2x^2$

Logo,

$det~A=3~.~det~B\\\\2x^2=3~.~(6x-2x^2)\\\\2x^2=18x-6x^2\\\\2x^2+6x^2=18x\\\\8x^2=18x\\\\4x^2=9x\\\\4x^2-9x=0\\\\x(4x-9)=0$

Isso será verdadeiro quando

$x=0$

e quando

$4x-9=0\\\\4x=9\\\\x=\dfrac{9}{4}\\\\x=2,25$

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