Invariant Lines of Matrix Transformation

So I am trying to find the invariant lines of a 2D matrix transformation of an enlargement by scale factor -3 about the origin. The question specifically is "Given that the line y = mx, where m is a real constant, is invariant under the transformation, find the two possible values of m". Textbook says the answer is m= +/- 1 but I have derived an answer of: m can take any real value. Could someone just verify whether the textbook is correct here as I have used a different method and the answer is not well explained in the textbook.

Thanks!

What's your matrix?

The top row of the 2x2 matrix is (-3,0) then the bottom row is (0,-3).

So I am trying to find the invariant lines of a 2D matrix transformation of an enlargement by scale factor -3 about the origin. The question specifically is "Given that the line y = mx, where m is a real constant, is invariant under the transformation, find the two possible values of m". Textbook says the answer is m= +/- 1 but I have derived an answer of: m can take any real value. Could someone just verify whether the textbook is correct here as I have used a different method and the answer is not well explained in the textbook.

Thanks!

So I am trying to find the invariant lines of a 2D matrix transformation of an enlargement by scale factor -3 about the origin. The question specifically is "Given that the line y = mx, where m is a real constant, is invariant under the transformation, find the two possible values of m". Textbook says the answer is m= +/- 1 but I have derived an answer of: m can take any real value. Could someone just verify whether the textbook is correct here as I have used a different method and the answer is not well explained in the textbook.

Thanks!

I think youre right and the textbook is wrong but here is my method:

if you call your matrix 'A' here
do A x (x)/(mx) = (x1)/(mx1)
Then you have two equations for the top and bottom

eqtn 1: -3x= x1
eqtn 2: -3mx= mx1

substitute eqtn 1 into eqtn 2 like so

-3mx=m(-3x)

as both sides equal each other, m can take any value as it is a constant. also I just noticed, as your matrix is an enlargement through the origin, it has an infinite amount of invariant lines! so you're definitely right

The textbook is not correct, you are.

See here