Matrix equation.

We define

A (r, \theta ) = \begin{pmatrix} r\cos \theta & r\sin \theta \\ -r\sin \theta & r\cos \theta \end{pmatrix}

and the question asks me to find all the values of

\theta

and

r

(r, \theta \in \mathbb{R}: r>0)

such that

\left[A(r,\theta)\right]^n = I

where I is the 2x2 identity matrix.

My thoughts thus far were to prove that

\left[A(r,\theta)\right]^n = A(r^n, n\theta)

by induction (which I managed) and then solve by equating elements of the LHS and RHS.

Doing that, I came down to

r^{2n}=1 \implies r^n=\pm 1

, where I get stuck because I don't know what n is (all I've been told that n is a positive integer).

I think I've rightly found

\theta = \dfrac{k}{n}\pi

for some integer k.

Any tips on how to determine r?

(edited 12 years ago)

I can't see what is the role of

r

in your definition of the matrix

A

I can't see what is the role of

r

in your definition of the matrix

A

My apologies, that was a LaTeX fail on my part. :p:

Edited OP.

I'm not sure why you get stuck; you've been told r>=0, so if r^n = +/-1 you must have r = 1.

I'm not sure why you get stuck; you've been told r>=0, so if r^n = +/-1 you must have r = 1.

How did I miss that... Thank you.

LaTex

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