Question

Considere a matriz A = (aij) 3x3 em que aij = $\left \{ {{x,se{ \to \ = j} \atop {1,se_{\to i \neq j }} \right.$ Determine os valores de x para quais det A =2.

Answer 1

Primeiro, vamos montar a matriz

$A= \left[\begin{array}{ccc}a_1_1&a_1_2&a_1_3\\a_2_1&a_2_2&a_2_3\\a_3_1&a_3_2&a_3_3\end{array}\right]$

$a_1_1=x\\ a_1_2=1\\ a_1_3=1\\ a_2_1=1\\ a_2_2=x\\ a_2_3=1\\ a_3_1=1\\ a_3_2=1\\ a_3_3=x$

A matriz ficou assim:

$A= \left[\begin{array}{ccc}x&1&1\\1&x&1\\1&1&x\end{array}\right]$

Agora, basta igualar a 2 e terminar:

$\\\\ \left[\begin{array}{ccc}x&1&1\\1&x&1\\1&1&x\end{array}\right] =2\\\\ x^3+1+1-x-x-x=2\\\\ x^3-3x+2=2\\\\ x^3-3x=0\\\\ x(x^2-3)=0\\\\ \boxed{x=0}\\\\ x^2-3=0\\\\ x^2=3\\\\ x=+\ ou\ -\ \sqrt{3}\\\\ \boxed{\therefore\ S(0,-\sqrt{3},\sqrt{3})}$

Os valores de x: 0,-√3 e √3