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

then prove that

- (A+B) (A-B) = A
^{2}-B^{2} - (A-B)
^{2}= A^{2}-2AB+B^{2} - (A+B)
^{3}= A^{3}+3A^{2}B+3AB^{2}+B^{3}

AB = BA--------(1)[Given]

i) (A + B)(A - B) = A^{2} - AB + BA - B^{2}.

or (A + B)(A - B) = A^{2} - BA + BA - B^{2}.[From(1)]

or (A +B)(A - B) = A^{2} - B^{2}.

ii) (A - B)^{2} = (A - B)(A - B) + A^{2} - AB - BA +B^{2}.

or (A - B)^{2} = A^{2} - AB - AB + B^{2} [From(1)]

or (A - B)^{2} = A^{2} - 2AB + B^{2}.

iii) (A + B)^{3} = A^{3} + 2A^{2}B + A^{2}B + 2AB^{2} + AB^{2} + B^{3}.

or (A + B)^{3} = A^{3} + 3A^{2}B + 3AB^{2} + B^{3}.