Matrices question. PLSSSSS HELP.

Question

4. Let n≥2, let A and B b en×n matrices and let O_n be the n×n matrix with every entry 0.

(b) Suppose that A² = O_n. Show that I_n − A is invertible, and that (I_n − A)^-1 = I_n + A.

(c) Suppose that A^m = O_n for some positive integer m. Show that I_n − A is invertible. 

Answer 1

 

 Given:   n>=2; A and b are square nxn matrices; 0n  is the nxn zero matrix;

 

  (b)      Given: A^2 = 0n;  

            Prove: I - A is invertible and it's inverse is I + A

       

               Proof:                

                    (I +A)(I-A) = I*I - I*A + I*A - A*A <--- FOIL method

                                     = I -     A   + A  - A*A  <--- properties of identity matrix

                                      = I  - A*A  <--- matrix A cancels out

                                       = I - 0  <---- A*A is given to be zero

                                       = I <--- property of zero matrix

 

  Likewise,

                   (I-A)(I+A) = I*I + i*A - I*A - A*A

                                   = I +A -A -A*A

                                    = I - A*A

                                      = I -0

                                      = I

  So I+A and I-A are inverses of each other

 

 

(C)    A^m = 0 for some positive integer m;

        

 

        ( I+A^m)(I - A^m) = I - A^m + A^m + A^(2m)

                                     = I - 0 + 0 + (A^m)^2

                                     = I + 0^2 = I

 

      But (I - A^m) can be factored into (I-A) P(A) where P(A) is a polynomial function of matrix A

     So the inverse of (1-A) is (1+A^m)*p(A) which shows it is invertible