# Thread: finding the 2 x 2 matrix

1. ## finding the 2 x 2 matrix

If I get a question in the exam that shows any transformation, and then asks me to find the "2 x 2 matrix which represents the transformation", how would I find it? Is this a case of memorizing specific matrices?

Here's a real question from a past paper:

(a) Describe fully the single transformation which maps triangle A onto triangle B.

Rotation, y = x

(b) Find the 2 × 2 matrix which represents this transformation.

2. It's actually REFLECTION about the line y = x.

What transformation matrix represents reflection?

3. That's the thing, I don't know.

4. You know that this particular reflection swaps the x and y coordinates, so:

you know from matrix multiplication that this means:
ax1 + bx2 = x2
=> a=0, b=1

cx1+dx2 = x1
=>c=1,d=0

so the transformation matrix is:

5. Originally Posted by SpringFan25
You know that this particular reflection swaps the x and y coordinates, so:
How do I know this?

you know from matrix multiplication that this means:
ax1 + bx2 = x2
=> a=0, b=1
I don't understand how you did this sum. How does ax1 + bx2 = x2 lead me to know the values of a and b?

6. A reflection about the line y = x is the same as finding an inverse function, in which the domain and range are swapped - in other words, the x and y values are swapped.

Surely if ax_1 + b_2 = x_2, the right hand side is 0x_1 + 1x_2...

7. Originally Posted by Prove It
A reflection about the line y = x is the same as finding an inverse function, in which the domain and range are swapped - in other words, the x and y values are swapped.
Okay thanks I get that now. :-)

Surely if ax_1 + b_2 = x_2, the right hand side is 0x_1 + 1x_2...
Sorry I must be the stupidest person alive, but I don't know what you mean by this! Could you explain it fuller? What right hand side are you talking about?

8. The right hand side of ax_1 + bx_2 = x_2...

Surely x_2 is the same as 0x_1 + 1x_2...

Therefore ax_1 + bx_2 = 0x_1 + 1x_2

What are a and b?

9. Oh ok, a = 0 and b = 1. So you basically just assigned two random values to a and b?

Sorry Im asking so many questions. I understand the basics of matrices (addition, multiplication, determinants, inverse) but transformational matrices seem to go over my head, and my textbook is terrible at explaining it.

10. No we did not assign two random values to a and b. For the equation to be balanced, the x_1 terms have to have the same coefficients, and the x_2 terms have to have the same coefficients.

So if ax_1 + bx_2 = x_2, there is clearly a zero x_1 term. So a = 0 and b = 1.

11. Ok thanks that makes perfect sense now! I think I've got it now. Thanks for your help. :-)

12. Is there a sort of guide online that you can point me to that gives a point by point instruction on transformations?

13. Wikipedia is a good place to start.

14. Ok perhaps I didn't understand it so well. Here's another sum I've got:

( 4.. is transformed into ( 8
..3 ) ............................6 )

So my workings are...
4a + 3b = 8

So now I'm stuck and I know I've missed something. Help, please!

15. What is the transformation doing? It's enlarging both the x and y values by a factor of 2. So clearly matrix is ..... Think about how the old x values are related to the new ones and the same with the y values.

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