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#1 2006-11-20 10:15:22

fangree01
Member
Registered: 2006-11-13
Posts: 11

Nonsingular Matrices

Hi,

I was wondering if anyone could help with these questions on matrices i can't get my head around them at all dizzy!!

Let A =     1 -2  -2 -2
               -2  1  -2 -2
               -2 -2   1 -2
               -2 -2  -2  1

Show that (3I - A)² is equal to a certain scalar multiple of 3I - A.  Hence prove that A is nonsingular and find A-¹.

And

Nonsingular nxn matrices A, B are given to satisfy (A + B)-¹ = A-¹ + B-¹.

Show that  C = -C-¹ - I where C = A-¹ B. Deduce that (A-¹ B)³ = I

We have not done the determinant yet so i cant use that i have the answer but i still dont get how to do it.  I would be grateful for any advice.

Thanks

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#2 2006-11-23 22:44:27

gnitsuk
Member
Registered: 2006-02-09
Posts: 121

Re: Nonsingular Matrices

The definition of a singular matrix is one which does not have an inverse. In order to determine wether a matrix has an inverse or not one has to evaluate it's determinant. The definition is simply:

A matrix is singular if and only if its determinant is 0

So given that you say you have not yet covered determinants how would you be expected to determine if a matrix is singular or not?

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