if AX= kX k belongs to real numbers

A=|4 -2| X=|x|
|-2 4| |y|

and x is not equal to cero, and cero is not equal to y, find k

To find the value of k in the equation AX = kX, we first need to find the eigenvalues of matrix A.

Let's start by finding the determinant of A - kI, where I is the identity matrix of the same size as A:

A - kI = |4 - 2k -2|
|-2 4 - 2k|

Taking the determinant of this matrix, we get:

det(A - kI) = (4 - 2k)(4 - 2k) - (-2)(-2)
= 16 - 8k - 8k + 4k^2 + 4
= 4k^2 - 16k + 20
= 4(k^2 - 4k + 5)

Now, set det(A - kI) equal to 0:

4(k^2 - 4k + 5) = 0

Divide both sides by 4:

k^2 - 4k + 5 = 0

This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula:

k = (-(-4) ± √((-4)^2 - 4(1)(5))) / (2(1))
k = (4 ± √(16 - 20)) / 2
k = (4 ± √(-4)) / 2

The discriminant is negative (-4), which means there are no real solutions for k. Therefore, there is no real value of k that satisfies the equation AX = kX.