Question

If $A$ and $B$ are two square matrices of the same order and $m$ is a positive integer, then $(A+B{)}^m$ = ${}^m{C}_o{+}^m{C}_1{A}^{m-1}B{+}^m{C}_2{A}^{m-2}{B}^2+....{+}^m{C}_m{B}^m,$ if

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

Answer 1

Answer: (D)

Given, $(A+B{)}^m={}^m{C}_0{A}^m+{}^m{C}_1{A}^{m-1}B$
$+{}^m{C}_2{A}^{m-2}{B}^2+...+{}^m{C}_m{B}^m$
Put $m=2$
$(A+B{)}^2={}^2{C}_0{A}^2+{}^2{C}_{1}AB+{}^2{C}_2{B}^2$
$\Rightarrow A^2+B^2+AB+BA=A^2+2AB+B^2$
$\Rightarrow AB+BA=2AB$
$\Rightarrow AB=BA$