22 votes 22 votes Consider the following matrix $$A = \left[\begin{array}{cc}2 & 3\\x & y \end{array}\right]$$ If the eigenvalues of A are $4$ and $8$, then $x = 4$, $y = 10$ $x = 5$, $y = 8$ $x = 3$, $y = 9$ $x = -4$, $y =10$ Linear Algebra gatecse-2010 linear-algebra eigen-value easy + – gatecse asked Sep 21, 2014 • edited Jun 7, 2018 by Milicevic3306 gatecse 8.6k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply mohan123 commented Nov 15, 2019 reply Follow Share 2y-3x=32 x=-4 , b= 10 1 votes 1 votes Hira Thakur commented Jan 24 reply Follow Share Similar question: GATE CSE 2015 Set 1 | Question: 36 0 votes 0 votes Please log in or register to add a comment.
Best answer 36 votes 36 votes Sum of eigenvalues is equal to trace (sum of diagonal elements) and product of eigen values is equal to the determinant of matrix So, $2+y=8+4$ and $2y-3x = 32$ Solving this we get $y = 10, x =-4.$ Option $D$ is answer. Pooja Palod answered Sep 26, 2015 • edited Jun 11, 2018 by Milicevic3306 Pooja Palod comment Share Follow See all 0 reply Please log in or register to add a comment.
4 votes 4 votes Solve the equation 3x 2y=8 and x 2y=16 which i get x= -4 ,y = 10 Bhagirathi answered Sep 21, 2014 Bhagirathi comment Share Follow See all 0 reply Please log in or register to add a comment.
1 votes 1 votes As we know The product of eigenvalue is equal to Determinant. So in this question, eigenvalues are given 4 and 8. So are determinant is 32. Now we can put the value of X and Y and check whether the determinant is equal to 32 or not ayushjain321 answered Oct 12, 2023 ayushjain321 comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes I found one more interesting solution, which will give you intuition as per @sachin sir language. Follow the steps: Select Option 1, make a matrix. Subtract eigenvalues from the main diagonal. Check if the matrix has linearly dependent columns. Follow the above 3 steps for other options as well. I know the solution provided by Pooja is faster but it is also an interesting way to think about the solution. me.himanshu.k answered Jan 27 me.himanshu.k comment Share Follow See all 0 reply Please log in or register to add a comment.