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Question

If 2,-4 are the eigen value of a non-singular matrix A and IAI=-8, then the eigen vaule of Adj A are x and -y then the value x+y is ?

Comments

Refer: https://math.stackexchange.com/questions/100161/eigenvalues-of-adjoint-of-non-singular-matrix

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Didn’t understood @adad20 :(

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@Chhaatra

We know

$Ax = λx$, where λ is eigen value and x is eigen vector

Multiply both side by $A^{-1}$ from left

= $A^{-1}Ax = A^{-1}λx$

= $x = A^{-1}λx$

= $A^{-1}x = \frac{1}{λ}x$

so we can conclude that if eigenvalues of matrix $A$ are λ1, λ2, …,λn the eigenvalues corrresponding to its inverse i.e. $A^{-1}$ will be $\frac{1}{λ1}, \frac{1}{λ2},…, \frac{1}{λn}$.

We know that $A^{-1} = \frac{adjA}{|A|}$ since matrix is non-singular and its inverse exists.

$adjA = A^{-1} |A|$

Therefore eigenvalues of $adjA$ will be $|A|$* eigenvalues corresponding to $A^{-1}$

i.e. $\frac{|A|}{λ1}, \frac{|A|}{λ2},…, \frac{|A|}{λn}$

 

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Thank you!

Answer 1

Eigen value of Adj A= det(A)/eigen value of (A)

now here ,

det(A)=-8

eigen value =2,-4

eigen value of Adj A= -8/2 =-4=x

                                =-8/-4=2=-y

x+y=-4-2=-6 [note -y=2 so y=-2]