Let A be a 2x2 matrix.
Then, prove the following:
This might work, but check my computations. Since , , so if we call the entries in A a,b,c,d, we get ad=bc. Now this is the part you need to check- it looks like using this gives a nice expression for any power of A, with entries involving powers of (a+d). I think this should do it.
In my previous post we can avoid the concept of rank : if with and then, etc. So, the problem can be solved using only the concepts of determinant and product of matrices, obtaining more simple algebra . I don't mean exactly linear algebra (some say Linear Algebra ends when Cramer's Rule starts) .
Isn't it "where determinants start"? I read an interesting paper lately: http://www.axler.net/DwD.pdf It was a revelation to me, a mediocre student who instantly falls asleep on seeing formulas with many indices.
As for the problem, do you think it's tractable by means of finding a formula for dependent on a, b, c and d? I've just tried it and I can't but I'm not very good.
Well, that depends on the way we categorize.
The characteristic polynomial for a matrix is . If we know that then, . Now, we can use the Cayley-Hamilton theorem. For example for you'll obtain .As for the problem, do you think it's tractable by means of finding a formula for dependent on a, b, c and d? I've just tried it and I can't but I'm not very good.