Prove the formula of inverse matrix

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May 20th 2011, 02:59 PM
mahefo

Prove the formula of inverse matrix

In econometric, I have to use this formula to compute the inverse matrix
$(A\pm bc')^{-1}=A^{-1}\mp \left[\frac{1}{1\pm c'A^{-1}b}\right]A^{-1}bc'A^{-1}$
where A is a $n\times n$ , 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.
May 20th 2011, 07:59 PM
FernandoRevilla

Have you tried

$(A\pm bc^t)\left(A^{-1}\mp \left(\frac{1}{1\pm c^tA^{-1}b}\right)A^{-1}bc^tA^{-1}\right)=\ldots=I\quad \textrm{?}$
May 21st 2011, 02:24 AM
mahefo

I can prove it. But I can not explain it naturally. I do not know this formula is discovered from where.
May 21st 2011, 09:46 AM
Deveno

it is a special case of the Binomial Inverse Theorem, see here
May 22nd 2011, 01:59 AM
mahefo

Quote:

Originally Posted by Deveno

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

Thank you so much.
May 22nd 2011, 02:20 AM
FernandoRevilla

Quote:

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

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