SOLUTION: Suppose that A is a square matrix and A4 = 0 (the zero matrix). Show that (10 points) (I − A) −1 = I + A + A^2 + A^3 .

Let  B  be the matrix   +  +  + .



We need to show that  (I-A)*B = I  and  B*((I-A) = I.



For the first equality, we have

    (I-A)*B =  =  = 

                                        =  -  =  = I.


    Here we used the fact that   = 0,  which is given.

    Thus, the property  (I-A)*B = I  is proven.



For the second equality, we have

    B*(I-A) =  =  -  = 

                                        =  -  =  = I.


    Again, here we used the fact that   = 0,  which is given.

    Thus, the property  B*(I-A) = I  is proven.