# Calculate the inverse of the matrix using power series.

• Oct 26th 2012, 01:29 PM
sfspitfire23
Calculate the inverse of the matrix using power series.
Calculate the inverse of [1, i/2 ; -1/3, 1] = I - [0, -i/2 ; 1/3, 0] using a power series.

I know that (I - A)^-1 = I + A + A^2 +A^3 but am unsure of how to calculate this without using a computer.

Thanks
• Oct 26th 2012, 02:37 PM
johnsomeone
Re: Calculate the inverse of the matrix using power series.
There are a few ways to do this. Typically you want to diagonalize any matrix that you're going to raising to powers, but here you can avoid that.

$\displaystyle \text{Let } A = \left(\begin{matrix}0 & -i/2 \\ 1/3 & 0 \end{matrix}\right). \text{ Then } A^2 = cI, \text{ where } c = \frac{-i}{6}.$

$\displaystyle \text{Thus }\left(\begin{matrix}1 & i/2 \\ -1/3 & 1 \end{matrix}\right)^{-1}$

$\displaystyle = (I - A )^{-1}$

$\displaystyle = I + A + A^2 + A^3 +...$

Spoiler:

$\displaystyle = I + A + cI + cA + c^2I + c^2A + c^3I + c^3A + ...$

$\displaystyle = (1+c+c^2+...)I + (1+c+c^2+...)A$

$\displaystyle = (1+c+c^2+...)(I + A)$

$\displaystyle = \frac{1}{1-c}\left(\begin{matrix}1 & -i/2 \\ 1/3 & 1 \end{matrix}\right).$

$\displaystyle \text{Thus }\left(\begin{matrix}1 & i/2 \\ -1/3 & 1 \end{matrix}\right)^{-1} = \frac{1}{1-\frac{-i}{6}}\left(\begin{matrix}1 & -i/2 \\ 1/3 & 1 \end{matrix}\right) = \frac{1}{6+i}\left(\begin{matrix}6 & -3i \\ 2 & 6 \end{matrix}\right) = \frac{6-i}{37}\left(\begin{matrix}6 & -3i \\ 2 & 6 \end{matrix}\right).$

• Oct 27th 2012, 07:15 AM
sfspitfire23
Re: Calculate the inverse of the matrix using power series.
Thanks. How did you calculate c?
• Oct 27th 2012, 09:59 AM
johnsomeone
Re: Calculate the inverse of the matrix using power series.
I just did the matrix multiplication (they're 2x2 matricies with half zeros, so that's about than 10 second of work). I didn't know ahead of time it would work out this way. If was only after I did the multiplication that I saw that the power series would thereby algebraically "collapse" the way it did.

As an aside, that's a kinda universal approach I have to math problems. When I'm given something general or abstract, I try to do (or apply it to) a few specific, more concrete examples - or in this case compute a term or two of a series. That often helps me "see" what's going on. Sometimes, like with this problem, I'll find something unexpected that will give me the key to how to solve the problem.