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Question

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 _________.

Comments

$x+y=-6?$

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@Lakshman Patel RJIT

can you explain ??

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@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 $-y=2\Rightarrow y=-2$

$\Rightarrow x + y =-4-2=-6$

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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

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@pream sagar

you did some mistake please check your comment.

answer should be $-6$

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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

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@ pream sagar

see my comment, I also take eigen value of adj(A) matrix.

why you take $x=2$ and $-y=4?$