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If $A = \begin {bmatrix} \ x & 4x\\ \ 2 & y \\ \end{bmatrix}$ has eigen values 5 &-1 ,then find values of x& y.
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By the property we know that , sum of diagonal elements = sum of eigen values

i.e. $x + y =5-1=4$ or $x + y =4 ………….(1)$

Also |A|= product of eigen values

i.e. $xy-8x =5×(-1) = -5………………….(2)$

Solving equations (1) & (2) by putting $y=4 – x$ from (1) in (2) we get $x = 1 , - 5$

When x=1 ,from y= 4 – x ,y=3 & when x= -5 , y=9