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Question 928086: True or False (justify your answer

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
(a)

A2B = (AA)B = A(AB) = A(BA) = (AB)A = (BA)A = B(AA) = BA2 

(b)

               A2 - 3A + I = 0

            A2 - (3I)A + I = 0

       A-1[A2 - (3I)A + I) = A-1(0)

 A-1A2 - A-1[(3I)A] + A-1I = 0

A-1(AA) - [A-1(3I)]A + A-1 = 0

A scalar multiple of the identity commutes with all other matrices.

(A-1A)A - [(3I)A-1]A + A-1 = 0

     IA - (3I)[A-1A] + A-1 = 0

           A - (3I)I + A-1 = 0

              A - 3I + A-1 = 0

                         A = 3I - A            
 
(d)



is the coefficient matrix of this system for some p,q,r:

%22%22=%22%22

  
 
if lambda=4, the system is

%22%22=%22%22

system%28-2x%5B1%5D=p%2C%0D%0A%0D%0Ax%5B2%5D-x%5B3%5D=q%2C%0D%0A%0D%0A0x%5B3%5D=r%29

This is inconsistent unless r=0

--------------------------

if lambda=6, the system is

%22%22=%22%22

system%280x%5B1%5D=p%2C%0D%0A%0D%0Ax%5B2%5D-x%5B3%5D=q%2C%0D%0A%0D%0A2x%5B3%5D=r%29

This is inconsistent unless p=0
  
Edwin