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

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Solution

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

$∴$ $(A + B)^{2}$ = (A + B).(A + B)

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

= $A^{2}$ + AB + AB + $B^{2}$ [ $∵$ AB = BA]

= $A^{2} + 2 A B + B^{2}$
Hence proved.

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