4 votes 4 votes The Eigen values of $A=\begin{bmatrix} a& 1& 0\\1 &a &1 \\0 &1 &a \end{bmatrix}$ are______ $a,a,a$ $0,a,2a$ $-a,2a,2a$ $a,a+\sqrt{2},a-\sqrt{2}$ Linear Algebra engineering-mathematics linear-algebra eigen-value + – Hirak asked Apr 25, 2019 edited Apr 25, 2019 by Lakshman Bhaiya Hirak 1.2k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply Lakshman Bhaiya commented Apr 25, 2019 i edited by Lakshman Bhaiya Apr 25, 2019 reply Follow Share 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 $1,1,1$ ------------------------------------------------------------------ Apply the above properties to the your question then you will get answer $(d).$ Here $\text{Sum of all eigen values = a+a+a=3a}$ and $\text{Product of all eigen values =|A|=$a^{3}$-2a}$ 2 votes 2 votes Hirak commented Apr 30, 2019 reply Follow Share Thanks.. :) I made a BAD silly mistake while finding the determinant... my bad.. ! Thanks a ton.. :) 0 votes 0 votes Please log in or register to add a comment.
Best answer 6 votes 6 votes Plz see the picture below for detailed answer. 1. Using properties to eliminate options: 2. Using general procedure: SuvasishDutta answered Apr 27, 2019 edited May 1, 2019 by SuvasishDutta SuvasishDutta comment Share Follow See all 6 Comments See all 6 6 Comments reply Show 3 previous comments SuvasishDutta commented May 1, 2019 reply Follow Share Thanks. You can apply properties which will save time. I have added it. 0 votes 0 votes Lakshman Bhaiya commented May 1, 2019 reply Follow Share @SuvasishDutta @Hirak Good explanation, and always believe in method rather than a shortcut. And property just saves time but for better understanding, the actual method is good enough. 0 votes 0 votes SuvasishDutta commented May 1, 2019 reply Follow Share Yes absolutely right @Lakshman Patel RJIT. 1 votes 1 votes Please log in or register to add a comment.