Question

If A and B are two square matrices of the same order and m is a positive integer, then (A + B)^m = ^mC₀A^m + ^mC₁A^m - 1B + ^mC₂A^{m - 2}B² + … + ^mC^{m - 1}AB^{m - 1} + ^mC_mB^m if

(A) AB = BA

(B) AB + BA = 0

(C) Am = 0, Bm = 0

(D) None of these

Answer 1

Answer is (A) AB = BA

If we consider m = 2

(A + B)² = (A + B) (A + B) = A² + AB + BA + B²

= A² + 2AB + B² (if AB = BA)

Similarly

(A + B)^m = ^mC₀A^m + ^mC₁A^m-1B + ^mC₂A^m-2B² +… + ^mC_mB^m

holds if AB = BA.