# GATE2010-29

2.9k views

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. $x = -4$, $y =10$

edited
0

2y-3x=32

x=-4 , b= 10

Sum of eigenvalues is equal to trace (sum of diagonal elements) and product of eighen 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.

edited
Solve the equation 3x 2y=8 and x 2y=16 which i get x= -4 ,y = 10
using one simple property...

The sum of eigen values is equal to the sum of the diagonal elements.

Given that the eigen values are 4 and 8, we have,

8+4 = y + 2

y = 10.

Now out of A and D, I don't find any difference in the options.
1 flag:
✌ Edit necessary (manish_pal_sunny)

## Related questions

1
3k views
The larger of the two eigenvalues of the matrix $\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ is _______.
Consider the matrix as given below. $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 0 & 0 & 3\end{bmatrix}$ Which one of the following options provides the CORRECT values of the eigenvalues of the matrix? $1, 4, 3$ $3, 7, 3$ $7, 3, 2$ $1, 2, 3$
What are the eigenvalues of the following $2\times 2$ matrix? $\left( \begin{array}{cc} 2 & -1\\ -4 & 5\end{array}\right)$ $-1$ and $1$ $1$ and $6$ $2$ and $5$ $4$ and $-1$
For the below question, one or more of the alternatives are correct. Write the code letter$(s)$ $a$, $b$, $c$, $d$ corresponding to the correct alternative$(s)$ in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. The ... $(0,0,\alpha)$ $(\alpha,0,0)$ $(0,0,1)$ $(0,\alpha,0)$