1. ## Inverses

Hi,

Why is it that if B = A^-1,

B^-1 = A

I understand that whatever you do to one side, you have to do the same for the other side, but I don't understand how (B^-1)(B^-1) = B

Thanks.

2. What is this in regards to? What field are we working in? Matricies?

3. Originally Posted by CSG18
Hi,

Why is it that if B = A^-1,

B^-1 = A

I understand that whatever you do to one side, you have to do the same for the other side, but I don't understand how (B^-1)(B^-1) = B

$\displaystyle B^{-1} = (A^{-1})^{-1} = A$.
Alternatively, note that $\displaystyle B B^{-1} = I$. But if $\displaystyle B = A^{-1}$ then $\displaystyle A^{-1} B^{-1} = I \Rightarrow A A^{-1} B^{-1} = A \Rightarrow I B^{-1} = A ....$
4. The definition of $\displaystyle X^{-1}$, in general, (matrices, elements of a group or field, etc.) is that $\displaystyle Y= X^{-1}$ if and only if $\displaystyle XY= I$ and $\displaystyle YX= I$ where I is the "multiplicative identity" for the algebraic structure. T
If you know that $\displaystyle B= A^{-1}$ then you know that $\displaystyle AB= I$ and [mathy]BA= I[/tex], by replacing X above with A and Y with B. But if we were to replace X with B and Y with A, we wold get exactly the same equations!
Therefore, $\displaystyle A= B^{-1}$.