If A and B are two equal ordered matrix such that AB = BA

then prove that

  1. (A+B) (A-B) = A2-B2
  2. (A-B)2= A2-2AB+B2
  3. (A+B)3= A3+3A2B+3AB2+B3

given : AB = BA
‚Äč1.
LHS of the given equation is:
(A+B)(A-B)=A2+BA-AB-B2=A2+AB-AB-B2=A2-B2
= RHS
2.
LHS of the given equation is:
(A-B)2=(A-B)(A-B)=A2-BA-AB+B2=A2-AB-AB+B2=A2-2AB+B2
= RHS
3.
LHS of the given equation is:
(A+B)3=(A+B)(A+B)2=(A+B)(A2+2AB+B2) [since AB=BA]=A3+2A2B+ABB+BAA+2BAB+B3=A3+2A2B+AB2+ABA+2ABB+B3 [since AB = BA]=A3+2A2B+AB2+AAB+2AB2+B3=A3+2A2B+3AB2+A2B+B3=A3+3A2B+3AB2+B3
= RHS
hope this helps you.
 

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