Question Number 129887 by zarawan last updated on 20/Jan/21
![Is the vector [(1),((−2)),(1) ]an eigen vector of [(3,6,7),(3,3,7),(5,6,(5 )) ]? if sp find the corresponding eigen value?](Q129887.png)
$${Is}\:{the}\:{vector}\:\begin{bmatrix}{\mathrm{1}}\\{−\mathrm{2}}\\{\mathrm{1}}\end{bmatrix}{an}\:{eigen}\:{vector}\:{of}\:\begin{bmatrix}{\mathrm{3}}&{\mathrm{6}}&{\mathrm{7}}\\{\mathrm{3}}&{\mathrm{3}}&{\mathrm{7}}\\{\mathrm{5}}&{\mathrm{6}}&{\mathrm{5}\:}\end{bmatrix}?\:{if}\:{sp}\:{find}\:{the}\:{corresponding}\:{eigen}\:{value}? \\ $$
Answered by Olaf last updated on 20/Jan/21
![AX = [(3,6,7),(3,3,7),(5,6,5) ] ((1),((−2)),(1) ) = (((−2)),(4),((−2)) ) = −2 ((1),((−2)),(1) ) = −2X ∃λ\ AX = λX ⇒ X is a eigen vector of A The corresponding eigen value is −2](Q129922.png)
$$\mathrm{AX}\:=\:\begin{bmatrix}{\mathrm{3}}&{\mathrm{6}}&{\mathrm{7}}\\{\mathrm{3}}&{\mathrm{3}}&{\mathrm{7}}\\{\mathrm{5}}&{\mathrm{6}}&{\mathrm{5}}\end{bmatrix}\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}\:=\:\begin{pmatrix}{−\mathrm{2}}\\{\mathrm{4}}\\{−\mathrm{2}}\end{pmatrix} \\ $$$$=\:−\mathrm{2}\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}\:=\:−\mathrm{2X} \\ $$$$\exists\lambda\backslash\:\mathrm{AX}\:=\:\lambda\mathrm{X} \\ $$$$\Rightarrow\:\mathrm{X}\:\mathrm{is}\:\mathrm{a}\:\mathrm{eigen}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{A} \\ $$$$\mathrm{The}\:\mathrm{corresponding}\:\mathrm{eigen}\:\mathrm{value}\:\mathrm{is}\:−\mathrm{2} \\ $$