Question

Determine os valores de u e r para os quais det(A−μI)=0 sendo A = 2 1 0 1 e I = 1 0 0 1 a matriz identidade

Answer 1

Olá

Temos que:

$A = \left[\begin{array}{ccc}2&1\\0&1\end{array}\right]$ e $I = \left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$

Daí, 

$A -uI = \left[\begin{array}{ccc}2&1\\0&1\end{array}\right] - u. \left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$
$A -uI = \left[\begin{array}{ccc}2&1\\0&1\end{array}\right] - \left[\begin{array}{ccc}u&0\\0&u\end{array}\right]$
$A -uI = \left[\begin{array}{ccc}2-u&1\\0&1-u\end{array}\right]$

Calculando o determinante e igualando a 0, temos que:

$(2-u)(1-u)=0$
$2 -2u-u+u^{2}=0$
$u^{2}-3u+2=0$

Utilizando Bháskara, temos que:

$u = \frac{-(-3) +- \sqrt{(-3)^{2} - 4.1.2} }{2.1}$
$u = \frac{3 +- \sqrt{9 -8} }{2}$
$u = \frac{3 +- 1}{2}$

$u' = \frac{3+1}{2} = \frac{4}{2} = 2$
$u" = \frac{3-1}{2} = \frac{2}{2} = 1$

Portanto, u = 1 ou u = 2