Question

Obtendo o autovalor associado a matriz A-) l 4 -2 l sobre l 5-7 l e ao autorvetor V = l 2 l sobre l 1 l teremos o resultado seguinte.
Assinale a alternativa correta :
a) Lambda = 3
b) Lambda =4
c) Lambda =2
d) Lambda =1

Answer 1

Temos a matriz

$\mathbf{A}=\left[ \begin{array}{cc} 4&-2\\ 5&-7 \end{array} \right ]$

e o autovetor

$\mathbf{v}=\left[ \begin{array}{cc} 2\\ 1 \end{array} \right ]$



O autovalor $\lambda$ procurado é tal que

$\mathbf{A}\mathbf{v}=\lambda\mathbf{v}\\ \\ \left[ \begin{array}{cc} 4&-2\\ 5&-7 \end{array} \right ] \left[ \begin{array}{cc} 2\\ 1 \end{array} \right ] =\lambda\left[ \begin{array}{cc} 2\\ 1 \end{array} \right ]\\ \\ \\ \left[ \begin{array}{cc} 4\cdot 2+(-2)\cdot 1\\ 5\cdot 2+(-7)\cdot 1 \end{array} \right ]=\lambda \left[ \begin{array}{cc} 2\\ 1 \end{array} \right]\\ \\ \\ \left[ \begin{array}{cc} 8-2\\ 10-7 \end{array} \right ]=\lambda \left[ \begin{array}{cc} 2\\ 1 \end{array} \right]\\ \\ \\ \left[ \begin{array}{cc} 6\\ 3 \end{array} \right ]=\lambda \left[ \begin{array}{cc} 2\\ 1 \end{array} \right]\\ \\ \\ \left[ \begin{array}{cc} 3\cdot 2\\ 3 \cdot 1 \end{array} \right ]=\lambda \left[ \begin{array}{cc} 2\\ 1 \end{array} \right]\\ \\ \\ 3\left[ \begin{array}{cc} 2\\ 1 \end{array} \right ]=\lambda \left[ \begin{array}{cc} 2\\ 1 \end{array} \right]\\ \\ \\ \Rightarrow\;\;\lambda=3$


Resposta: alternativa $\text{a) }\lambda=3.$