What is the simplified inverse formula for A + B?

[Tedjn]

Hi everyone. This comes from Putnam and Beyond #82 paraphrased:

A and B are two n by n matrices such that AB = BA. There exist positive integers p and q such that A^p = I and B^q = 0. Find the inverse of A + B.

There is an easy way to construct an inverse. Start with (A+B)(A-B) = A² - B². Multiply by A² + B² to get A⁴ - B⁴. Repeat until the B term disappears, and then multiply by the appropriate power of A.

I couldn't see it in the time I spent on this problem, but there seems to be the potential for a much simpler inverse formula, since

1) this is in the algebraic identity portion of the book
2) there are multiple equivalent ways of writing this inverse

I'd be interested if someone here finds one :-)

 

[tiny-tim]

Hi Tedjn!

A and B are two n by n matrices such that AB = BA. There exist positive integers p and q such that A^p = I and B^q = 0. Find the inverse of A + B.

There is an easy way to construct an inverse. Start with (A+B)(A-B) = A² - B². Multiply by A² + B² to get A⁴ - B⁴. Repeat until the B term disappears …

But if q isn't a power of 2 … ?

Try a homogenous combination, of the form ∑_i a_iAⁱB^n-i

 

[Hurkyl]

Since A and B commute, I would have just used elementary calculus to expand (A+B)^{-1} -- particularly since B is nilpotent and thus acts as if it were "small".

The work involved in this solution is nothing more than simple algebra, I just would have spotted the method in the way I described above.