Math Is Fun Forum

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Rational Canonical Form

You may look up the title of this post if you wish, but it isn't important.  I had the following on an algebra homework:

Find a matrix A with rational entries such that A is not the identity matrix and A^3 = I.

Now I knew I was supposed to use the rational canonical form, but being lazy, the question seemed easy enough to solve by just multiplying everything out.  However, I quickly found this is not the case.

What I wish to see is that if anyone can solve the question using a reasoned method.  In other words, guess and check is not allowed.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."