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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

  1. $x = 4$, $y = 10$
  2. $x = 5$, $y = 8$
  3. $x = 3$, $y = 9$
  4. $x = -4$, $y =10$
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4 Answers

Best answer
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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.
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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

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I found one more interesting solution, which will give you intuition as per @sachin sir language.

Follow the steps:

  1. Select Option 1, make a matrix. 
  2. Subtract eigenvalues from the main diagonal.
  3. Check if the matrix has linearly dependent columns.
  4. 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.

Answer:

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