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# transformations and matrices P6 Q watch

1. Given that M =

4 -5
6 -9

and that the eigenvalues of M are 1 and -6

A transformation T:R^2 -> R^2 is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T.

How do I find a cartesian equation of the line

The mark scheme says “stating, implying or showing λ = 1 associated with point invarient line). Why is this? Why shouldn’t λ = -6...............

??
2. You want Mv = v, which means the eigenvalue (the coefficient of v on the RHS) should be 1.
3. It might help to think in the following way:
The line has equation y=kx (as it passes through the origin. Hence every point on the line has the general coordinate (x,kx).
If every point is mapped to itself then we have:

That is:

Dividing both sides by x:

Clearly so the line has equation

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