Math Is Fun Forum

Shelled

Member

Registered: 2014-04-15

Posts: 44

Matrix inverse proof

Let A be a n*n matrix and

Hi, I'm having trouble with the above question. I've made a start but I'm not sure if I'm approaching it correctly.
First of, there's a hint that suggests that I first consider the product AX. But if I do that, the dimensions of A and X are different so I would need to find the transpose of X?
so is it safe to assume

?

To find if A is invertible can I also do the following?
AX=XB (where B is some solution of A)

and for the formula:

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrix inverse proof

Hi;

First and second line.

(where B is some solution of A)

A is an n x n matrix and B is n x 1 column vector... so how can they be equal?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Matrix inverse proof

AX =  X diag( 入i )

V := X^(-1)

AX diag( 1/ 入i )V  =  X diag( 入i )diag( 1/ 入i ) V  = I

thus

A^(-1) =  X diag( 1/ 入i ) X^(-1)

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X'(y-Xβ)=0