# Thread: Prove the formula of inverse matrix

1. ## Prove the formula of inverse matrix

In econometric, I have to use this formula to compute the inverse matrix
$\displaystyle (A\pm bc')^{-1}=A^{-1}\mp \left[\frac{1}{1\pm c'A^{-1}b}\right]A^{-1}bc'A^{-1}$
where A is a $\displaystyle n\times n$, symmetric, nonsingular matrix; b,c are n-column vectors, c' is transpose of c.
I used it but I did know how to prove this formula. I need your help.

2. Have you tried

$\displaystyle (A\pm bc^t)\left(A^{-1}\mp \left(\frac{1}{1\pm c^tA^{-1}b}\right)A^{-1}bc^tA^{-1}\right)=\ldots=I\quad \textrm{?}$

3. I can prove it. But I can not explain it naturally. I do not know this formula is discovered from where.

4. it is a special case of the Binomial Inverse Theorem, see here

5. Originally Posted by Deveno
it is a special case of the Binomial Inverse Theorem, see here
Thank you so much.

6. Originally Posted by mahefo
I can prove it. But I can not explain it naturally. I do not know this formula is discovered from where.

Given $\displaystyle A,B$ matrices, the natural way to prove the formula $\displaystyle A^{-1}=B$ (that was your original question) is to prove $\displaystyle AB=I$ . Another question is : given $\displaystyle A$ , find $\displaystyle A^{-1}$ .