if AX = kX, k belongs to reals where A=|4 -2| <-(is a matrix) X=|x|<-(matrix)
|-2 4| |y|
and x is not equal to cero, cero is not equal to y, find k
To find the value of k in the equation AX = kX, where A and X are matrices, you need to perform matrix operations.
First, let's write the given equation in matrix form:
A * X = k * X
Now, perform matrix multiplication between A and X:
|4 -2| * |x| = k * |x|
|-2 4| |y| |y|
Multiplying the matrices gives:
|4x - 2y| = |kx|
|-2x + 4y| |ky|
Now, equate the corresponding elements of both sides of the equation:
4x - 2y = kx (Equation 1)
-2x + 4y = ky (Equation 2)
From Equation 1, isolate x:
4x - kx = 2y
x(4 - k) = 2y
x = (2y)/(4 - k)
From Equation 2, isolate y:
-2x + ky = 4y
4y + 2x = ky
y(4 + 2x) = ky
4 + 2x = k
x = (k - 4)/(2y)
Since we know that x is not equal to zero, we can substitute the expressions for x from the equations above:
(k - 4)/(2y) ≠ 0
k - 4 ≠ 0
k ≠ 4
Therefore, the value of k can be any real number except 4.