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!
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(edited 3 years ago)
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Original post by Zebedi1
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!

The top row of the 2x2 matrix is (-3,0) then the bottom row is (0,-3).
Original post by Zebedi1
The top row of the 2x2 matrix is (-3,0) then the bottom row is (0,-3).

Ok, call this matrix A. What equation are you working with?
Original post by Zebedi1
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
Original post by Zebedi1
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!

The textbook is not correct, you are.

See here
Original post by nevafradd
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

Thank you very much!
Original post by ghostwalker
The textbook is not correct, you are.

See here

Ahh gosh I can't believe I didn't find this before I posted this thread. Thanks for your help
Yep, I spent a while on this one, too.
Here's the correct answer, and also, the correct question.
Any line through the origin is invariant under enlargements (e.g. "under T"), so m can take any real value.
They MEANT to ask the two possible values of m given the y=mx is invariant "under P", P being the 2x2 matrix:
(0 -3)
(-3 0).
Then you get the textbook answers.
Note that P is a reflected enlargement (factor 3) in the line y = -x, so it makes sense that the invariant lines are y = x and y = -x (i.e. m = +1 or -1)
The original question is Mixed Exercise 7, question 3c of Edexcel Core 1 A-level textbook.
(edited 11 months ago)