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A transformation matrix is given by M= [4 -1]
[2 3]
Show that the transformation matrix has no invariant line using the equation of line y=mx
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Had the invariant line exist, the given matrix would have real eigenvalue.
But it is easy to show that all its eigenvalues are complex numbers, not real.
Indeed, the characteristic matrix is .
Its characteristic equation is
(4-x)*(3-x) + 2 = 0,
or
12 - 3x - 4x + x^2 + 2 = 0
x^2 - 7x + 14 = 0. (*)
The discriminant of this quadratic equation is d = (-7)^2 - 4*1*14 = 49 - 56 = -7 less than zero.
It means that the characteristic equation has no real solution/solutions;
so, the original matrix has no real eigenvalues.
Thus the statement is proved.
Solved.
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For your better understanding
Since this matrix has complex value eigenvalues, the associated transformation of a plane
is stretching-compression combined with rotation.
Therefore, there is no invariant lines: rotation excludes such lines.