Question

If $ A $ and $ B $ are two square matrices of the same order and $ m $ is a positive integer, then $ (A + B)^m $ = $ ^mC_o+^mC_1A^{m-1}B+^mC_2A^{m-2}B^2+....+^mC_mB^m, $ if

• A. $ AB = - BA $
• B. $ A^m = 0, B^m = 0 $
• C. $ AB = 2BA $
• D. $ AB = BA $

Answer 1

Answer: (D)

Given, $(A + B)^m = \,{}^mC_0 A^m + \,{}^mC_1 A^{m-1} B$
$+ \,{}^mC_2A ^{m-2}B^2 +...+ \,{}^mC_mB^m$
Put $ m = 2$
$( A + B )^2 = \,{}^2 C_0 A^2 + \,{}^2C_1 AB + \,{}^2C_2 B^2$
$\Rightarrow A^2 + B^2 + AB + BA = A^2 + 2AB + B^2$
$\Rightarrow AB + BA = 2AB$
$\Rightarrow AB = BA$