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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)
Reply 2
bump
Reply 3
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!

What's your matrix?
Reply 4
What's your matrix?

The top row of the 2x2 matrix is (-3,0) then the bottom row is (0,-3).
Reply 5
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:smile:
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
Reply 8
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:smile:

Thank you very much!
Reply 9
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
Reply 10
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)

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