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!

Thanks!

..

(edited 3 years ago)

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!

Thanks!

What's your matrix?

Original post by vc94

What's your matrix?

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!

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

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

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

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.

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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