1 votes 1 votes If 2, – 4 are the eigen values of a non-singular matrix A and |A| = –8, then the eigen values of Adj A are x and –y then the value of x + y is _________. pream sagar asked Jan 7, 2019 pream sagar 1.1k views answer comment Share Follow See all 7 Comments See all 7 7 Comments reply Lakshman Bhaiya commented Jan 7, 2019 i edited by Lakshman Bhaiya Jan 7, 2019 reply Follow Share $x+y=-6?$ 0 votes 0 votes Magma commented Jan 7, 2019 reply Follow Share @Lakshman Patel RJIT can you explain ?? 0 votes 0 votes Lakshman Bhaiya commented Jan 7, 2019 reply Follow Share @Magma We can use the properties of Eigen values Important properties of Eigen values:- $(1)$Sum of all eigen values$=$Sum of leading diagonal(principle diagonal) elements=Trace of the matrix. $(2)$ Product of all Eigen values$=Det(A)=|A|$ $(3)$ Any square diagonal(lower triangular or upper triangular) matrix eigen values are leading diagonal (principle diagonal)elements itself. Example$:$$A=\begin{bmatrix} 1& 0& 0\\ 0&1 &0 \\ 0& 0& 1\end{bmatrix}$ Diagonal matrix Eigenvalues are $1,1,1$ $B=\begin{bmatrix} 1& 9& 6\\ 0&1 &12 \\ 0& 0& 1\end{bmatrix}$ Upper triangular matrix Eigenvalues are $1,1,1$ $C=\begin{bmatrix} 1& 0& 0\\ 8&1 &0 \\ 2& 3& 1\end{bmatrix}$ Lower triangular matrix Eigenvalues are $1,1,1$ Now, $A=\begin{bmatrix} 2 &0 \\ 0 &-4 \end{bmatrix}$ $Adj(A)=\begin{bmatrix} -4 &0 \\ 0 &2 \end{bmatrix}$ here $x=-4$ and $-y=2\Rightarrow y=-2$ $\Rightarrow x + y =-4-2=-6$ 2 votes 2 votes pream sagar commented Jan 7, 2019 reply Follow Share i think ans should be +6 if x is eigen value of Matrix A Then $\frac{|A|}{x}$ is eigen value of Adj(A) now come to question 2, – 4 are the eigen values of a non-singular matrix A and |A| = –8 Then eigen value of Adj(A) $\frac{- 8}{2}$ and $\frac{-8}{-4}$ i.e. -4 and 2 respectively given that x and -y are eigen values then x=2 and y=4 hence x+y =6 please correct me if i wrong 0 votes 0 votes Lakshman Bhaiya commented Jan 7, 2019 reply Follow Share @pream sagar you did some mistake please check your comment. answer should be $-6$ 0 votes 0 votes pream sagar commented Jan 9, 2019 reply Follow Share actually you have taken x=−4 and −y=2⇒y=−2 ⇒x+y=−4−2=−6 and i have taken x=2 and -y=-4 ==>y=4 as they mention in question x and -y are eigen values of Adj matrix then x+y=6 0 votes 0 votes Lakshman Bhaiya commented Jan 9, 2019 i edited by Lakshman Bhaiya Jan 9, 2019 reply Follow Share @ pream sagar see my comment, I also take eigen value of adj(A) matrix. why you take $x=2$ and $-y=4?$ 0 votes 0 votes Please log in or register to add a comment.