Question

Encontre os autovalores destas duas questões. PRECISO COM URGÊNCIA!!

Answer 1

Resposta:

$\lambda_{1} = -2\\\lambda_{2} = 6$

$\bold{v}_{\lambda1} = (y,y)$

$\bold{v}_{\lambda2} = (-y,y)$

Explicação passo a passo:

$det\left[\begin{array}{ccc}2-\lambda&-4\\-4&2-\lambda\end{array}\right] =0$

Autovalores:

$(2-\lambda)^2-16=0\\\\\lambda_{1} = -2\\\lambda_{2} = 6$

Autovetores:

Para $\lambda_1$ = -2;  v = $\left[\begin{array}{ccc}x\\y\end{array}\right]$

$\left[\begin{array}{ccc}2-(-2)&-4\\-4&2-(-2)\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}0\\0\end{array}\right]$

$\left \{ {{4x-4y=0} \atop {-4x+4y=0}} \right. \\\\x=y$

Então, $\bold{v}_{\lambda1} = (y,y)$ sendo um de seus representantes o vetor $\bold{v}_1 = (1,1)$

Para $\lambda_2$ = 6;  v = $\left[\begin{array}{ccc}x\\y\end{array}\right]$

$\left[\begin{array}{ccc}2-6&-4\\-4&2-6\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}0\\0\end{array}\right]$

$\left \{ {{-4x-4y=0} \atop {-4x-4y=0}} \right. \\\\x=-y$

Então, $\bold{v}_{\lambda2} = (-y,y)$ sendo um de seus representantes o vetor $\bold{v}_2 = (-1,1)$