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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-




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?
$${Is}\:{the}\:{vector}\:\begin{bmatrix}{\mathrm{1}}\\{−\mathrm{2}}\\{\mathrm{1}}\end{bmatrix}{an}\:{eigen}\:{vector}\:{of}\:\begin{bmatrix}{\mathrm{3}}&{\mathrm{6}}&{\mathrm{7}}\\{\mathrm{3}}&{\mathrm{3}}&{\mathrm{7}}\\{\mathrm{5}}&{\mathrm{6}}&{\mathrm{5}\:}\end{bmatrix}?\:{if}\:{sp}\:{find}\:{the}\:{corresponding}\:{eigen}\:{value}? \\ $$
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
$$\mathrm{AX}\:=\:\begin{bmatrix}{\mathrm{3}}&{\mathrm{6}}&{\mathrm{7}}\\{\mathrm{3}}&{\mathrm{3}}&{\mathrm{7}}\\{\mathrm{5}}&{\mathrm{6}}&{\mathrm{5}}\end{bmatrix}\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}\:=\:\begin{pmatrix}{−\mathrm{2}}\\{\mathrm{4}}\\{−\mathrm{2}}\end{pmatrix} \\ $$$$=\:−\mathrm{2}\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}\:=\:−\mathrm{2X} \\ $$$$\exists\lambda\backslash\:\mathrm{AX}\:=\:\lambda\mathrm{X} \\ $$$$\Rightarrow\:\mathrm{X}\:\mathrm{is}\:\mathrm{a}\:\mathrm{eigen}\:\mathrm{vector}\:\mathrm{of}\:\mathrm{A} \\ $$$$\mathrm{The}\:\mathrm{corresponding}\:\mathrm{eigen}\:\mathrm{value}\:\mathrm{is}\:−\mathrm{2} \\ $$

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