Further Pure 1

I'm completely stuck on question 2- I've managed the whole rest of the exercise but thins has got me stuck!

All I manage to to is create two formulas with about 4 unknowns in each and I can't see how to simplify it to make it a proof!

Thank you you for any help!

I'm completely stuck on question 2- I've managed the whole rest of the exercise but thins has got me stuck!

All I manage to to is create two formulas with about 4 unknowns in each and I can't see how to simplify it to make it a proof!

Thank you you for any help!

I'm completely stuck on question 2- I've managed the whole rest of the exercise but thins has got me stuck!

All I manage to to is create two formulas with about 4 unknowns in each and I can't see how to simplify it to make it a proof!

Thank you you for any help!

Have you made any further progress on this?

I just tried it purely from the equations and it drops out quite nicely!

Presumably you can do the matrix multiplication OK, so you know that T takes (x y) to (ax + cy, bx + dy). An invariant point maps to itself so we need

ax + cy = x
bx + dy = y

I would rearrange each equation so that it looks like
mx = ny for some constants m and n and then multiply one of the equations by the multiplier in the other equation and substitute back.

You should then end up with an equation that looks like
px = qx for some constants p and q

If you want a solution that isn't the origin (0, 0) then you can set p = q which should reduce to a condition on a, b, c and d. Substitute this into your determinant and simplify and the result comes out.

Post your working if still stuck

Thank you! I'm still stuck though!

I have attached a picture of what I have so far but I just can't see how to go any further!

I also don't understand the significance of being told detT = a+d-1
I fell like this is an important part because they wouldn't have told you that if it wasn't important but I don't see how to use it!

Sorry for all the questions but I'm self teaching myself the course!

Thank you

Thank you! I'm still stuck though!

I have attached a picture of what I have so far but I just can't see how to go any further!

I also don't understand the significance of being told detT = a+d-1
I fell like this is an important part because they wouldn't have told you that if it wasn't important but I don't see how to use it!

Sorry for all the questions but I'm self teaching myself the course!

Thank you

You're trying to show that you need det T = a + d - 1 in order to have some invariant points that are different from the origin (0 0) so you can't assume it to start with.

As you've correctly shown,
(a - 1)x = -cy
bx = (1 - d)y

If you multiply the 2nd of these equations by c you get

bcx = c(1 - d)y = (1 - d)(cy) = (1 - d)(1 - a)x

(replacing cy from the 1st equation we wrote down)

Now, x = 0 will always be a solution to this, but implies that y = 0 too so all we get is the origin as an invariant point! If we want some other points to be invariant then we can divide both sides of our equation by x to get

bc = (1 - d)(1 - a) (*)

Now, what is the determinant of the matrix they give you, and what happens if you put equation (*) into the determinant?

Thank you for that help - it was brilliant!

I have now done the calculation and got this:



Am I right in thinking that because when I rearrange the determinant (the bottom section of my working) that it equals the simultaneous equations that I have shown that an invariant point exists other than the origin?

And that because I divided the simultaneous equations by x I've shown it's not the origin?

I hope that makes sense!

Thank you!