Inverse of rank-one perturbation of identity

__Problem statement:__

If and are m-vectors, the matrix is known as a rank-one perturbation of the identity. Show that if is nonsingular, then its inverse has the form

for some scalar .

- give an expression for

- for what and is singular?

- if singular, what is

__Attempt:__

*Expression for :*

if

I did not actually show that the inverse has that form, and I'm not sure how to do so.

__For what and is singular:__

A is singular when is a unit vector on one of the axis, and is a unit vector in the opposite direction of .

Ex.

__If singular, what is :__

has rank m, and has rank 1.

If is singular has rank .

has then one free variable and its nullspace has rank 1.

It is a line in

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Just throwing around some ideas on the last two questions as I hate to post a question without attempt.

It seems that I can not get any further by my self,so I'm hoping some of you might spare a few moments.

Thanks.