SOLUTION: If A is a square matrix with the property (A with an exponet of 2) = 0, show that (I - A)inverse = A + I. Thank you

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Question 87107This question is from textbook
: If A is a square matrix with the property (A with an exponet of 2) = 0, show that (I - A)inverse = A + I.

Thank you This question is from textbook

Found 2 solutions by longjonsilver, Edwin McCravy:
Answer by longjonsilver(2297)   (Show Source): You can put this solution on YOUR website!
Any matrix multiplied by its inverse is equal to the identity matrix, I by definition. So:



Post multiply both sides by (I+A):


now,





So, becomes


and hence

cheers
Jon
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
If A is a square matrix with the property (A with an exponet of 2) = 0, show
that (I - A)inverse = A + I. 
The best way to get insight into a problem like this is first (ON SCRATCH PAPER)
to assume it is true and work backwards until you run into something that is
true.  Then write the steps in reverse and discard the scratch paper.

On scratch paper (NOT TO TURN IN), assume the proposition is already known to
be true (which of course it isn't, which is why in the end you must discard the
scratch work).

SCRATCH WORK:



Right multiply both sides by 



The left side is the identity .
And since matrix multiplication is distributive,
we may use "FOIL" to multiply out the right side:



Now we see on the right that , making
the  and the  cancel.  



Furthermore , so we have



and since we are given that 





and we are left with



Now this is equal, so let's turn everything backwards:

                     The identity matrix equals itself
                 Adding the  matrix to the right side 
                Replacing the  matrix by  
               Replacing  by  (they are given equal)
              Replacing  by  
       Adding and subtracting  on the right side
     Replacing the first  by  and the second by 
     Writing  as  and  as 
   Matrix multiplication is distributive over
                          matrix addition.
        Matrix multiplication is distributive over
                          matrix addition.
 Replacing the  on the left by the inverse of a matrix times the matrix.

[] = []  Right multiply both sides by 

[] = [] Matrix multiplication is associative.

 =   A matrix times its inverse is 

 =   Property of the identity matrix 

Be sure to discard the scratch work.

Edwin
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