0 votes 0 votes 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 ? Sagar475 asked Jan 15, 2022 Sagar475 286 views answer comment Share Follow See all 4 Comments See all 4 4 Comments reply Shoto commented Jan 15, 2022 reply Follow Share Refer: https://math.stackexchange.com/questions/100161/eigenvalues-of-adjoint-of-non-singular-matrix 2 votes 2 votes Chhaatra commented Jan 15, 2022 reply Follow Share Didn’t understood @adad20 :( 0 votes 0 votes Shoto commented Jan 15, 2022 reply Follow Share @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. $\frac{|A|}{λ1}, \frac{|A|}{λ2},…, \frac{|A|}{λn}$ 2 votes 2 votes Chhaatra commented Jan 15, 2022 reply Follow Share Thank you! 1 votes 1 votes Please log in or register to add a comment.
3 votes 3 votes 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] Kabir5454 answered Jan 15, 2022 Kabir5454 comment Share Follow See all 0 reply Please log in or register to add a comment.