Question

Show that if A and B are square matrices such that AB = BA, then $(A+B{)}^2=A^2+2AB+B^2$.

Answer 1

Since A and B are square matrices such that AB = BA,

$\therefore$ $(A+B{)}^2$ = (A + B).(A + B)

$=A^2$ + AB + BA + $B^2$

=$A^2$ + AB + AB + $B^2$ [$\because$ AB = BA]

= $A^2+2AB+B^2$
Hence proved.