If A and B are two equal ordered matrix such that AB = BA
then prove that
- (A+B) (A-B) = A2-B2
- (A-B)2 = A2-2AB+B2
- (A+B)3 = A3+3A2B+3AB2+B3
AB = BA--------(1)[Given]
i) (A + B)(A - B) = A2 - AB + BA - B2.
or (A + B)(A - B) = A2 - BA + BA - B2.[From(1)]
or (A +B)(A - B) = A2 - B2.
ii) (A - B)2 = (A - B)(A - B) + A2 - AB - BA +B2.
or (A - B)2 = A2 - AB - AB + B2 [From(1)]
or (A - B)2 = A2 - 2AB + B2.
iii) (A + B)3 = A3 + 2A2B + A2B + 2AB2 + AB2 + B3.
or (A + B)3 = A3 + 3A2B + 3AB2 + B3.