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# Matrix Doubt watch

1. [2 0
0 2 ] =M

[0 -1
1 0 ] =L

[1 -1
1 1 ] =N

a) Explain the geometrical effect of the transformations L and M.
b) Show that LM = N^2

The transformation represented by the matrix N consists of a rotation of angle theta about O, followed by an enlargement, center O, with a positive scale factor k
c) Find the value of theta and k

Firstly, for part a, I know that the geometrical effect of L is a rotation of 45 degrees counterclockwise around 0,0.
But could it also be seen as a switching of the x and y co-ordinate and then a reflection of that in the x - axis? ( because when I multiply M with [x,y] , i get the transformation [-y,x]

And, suppose we had LM, would it mean that the matrix transformation of M is being applied first or is it L thats being applied first?
Lastly, how do I go about c). Can someone explain?

2. (Original post by confuzzled92)
[2 0
0 2 ] =M

[0 -1
1 0 ] =L

[1 -1
1 1 ] =N

a) Explain the geometrical effect of the transformations L and M.
b) Show that LM = N^2

The transformation represented by the matrix N consists of a rotation of angle theta about O, followed by an enlargement, center O, with a positive scale factor k
c) Find the value of theta and k

Firstly, for part a, I know that the geometrical effect of L is a rotation of 45 degrees counterclockwise around 0,0.
But could it also be seen as a switching of the x and y co-ordinate and then a reflection of that in the x - axis? ( because when I multiply M with [x,y] , i get the transformation [-y,x]
THE effect of L is rotation of 90 degrees counterclockwise around origin.
Yes, you get this by switching of x and y coordinates then change the sign of one of them. If you change the sign of switched first coordinate then you will rotate
counterclockwise, and if you change the sign of the other then you will rotate clockwise.
And, suppose we had LM, would it mean that the matrix transformation of M is being applied first or is it L thats being applied first?
Lastly, how do I go about c). Can someone explain?

You have to apply L first, because generally the matrix multiplication is not commutitative (despite that here it is)
3. (Original post by ztibor)

You have to apply L first, because generally the matrix multiplication is not commutitative (despite that here it is)
Generally, I would have thought M first; as in LMx = L(Mx)
4. i thought M is applied first too but solutionbank assumes that L is applied first.

But how do I work out c?
5. (Original post by confuzzled92)
i thought M is applied first too but solutionbank assumes that L is applied first.
I was looking at the part b).

For part c) L is the rotation matrix and the order is actually the reverse of what you have in part b), and as ztitor said they commute in this case.

But how do I work out c?
Since a rotation doesn't effect an enlargement and vice versa, and one of ML produces the rotation and the other the enlargement, then you know what the rotation and the enlargement of N^2 is.

To find out what the rotation and enlargement of N is, note that rotations are additive, and enlargements are multplicative.

I.e. If the enlargement due to N is k, then the enlargment due to N^2 will be k times k....

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Updated: March 27, 2011
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