# Math Help - Matrix inversion problem

1. ## Matrix inversion problem

I've A^-1 = - A What is A
I've figured that A^2= -I but what's then?

2. ## Re: Matrix inversion problem

Originally Posted by ahmedzoro10
I've A^-1 = - A What is A
I've figured that A^2= -I but what's then?
$\mathbf{A}=\pm i\mathbf{I}$, where $i=\sqrt{-1}$.

3. ## Re: Matrix inversion problem

Originally Posted by alexmahone
$\mathbf{A}=\pm i\mathbf{I}$, where $i=\sqrt{-1}$.
Not necessarily. There can be many matrices whose square is –I. For 2x2 matrices, you can for example take $A = \begin{bmatrix}0&1\\ -1&0 \end{bmatrix}.$

4. ## Re: Matrix inversion problem

How did you get this one?

5. ## Re: Matrix inversion problem

Originally Posted by ahmedzoro10
How did you get this one?
Just note that $\sqrt{\mathbf{I}}=\mathbf{I}$ (there are others as well) and $\sqrt{-1}=\pm i$.

6. ## Re: Matrix inversion problem

Originally Posted by Opalg
Not necessarily. There can be many matrices whose square is –I. For 2x2 matrices, you can for example take $A = \begin{bmatrix}0&1\\ -1&0 \end{bmatrix}.$
So how many solutions does a matrix equation like $\mathbf{A}^2=-\mathbf{I}$ have? (where $\mathbf{A}$ and $\mathbf{I}$ are n by n matrices.)

7. ## Re: Matrix inversion problem

Let $A= \begin{bmatrix}a & b \\ c & d\end{bmatrix}$. Then $A^2= \begin{bmatrix}a^2+ bd & ab+ bd \\ ac+ cd & bc+ d^2\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. So we must have $a^2+ bd= 1$, $ab+ bd= 0$, $ac+ cd= 0$, and $bc+ d^2= 1$. You might try things like taking b (or c) equal to 0 or not equal to 0 and see what you get.