Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.

Home » Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.

Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.

Given,

$A\text{ }and\text{ }B$

are square matrices with the end goal that

$AB\text{ }=\text{ }BA.$

In this way,

${{\left( A\text{ }+\text{ }B \right)}^{2}}~=\text{ }\left( A\text{ }+\text{ }B \right)\text{ }.\text{ }\left( A\text{ }+\text{ }B \right)$

$=\text{ }{{A}^{2}}~+\text{ }AB\text{ }+\text{ }BA\text{ }+\text{ }{{B}^{2}}$

$=\text{ }{{A}^{2}}~+\text{ }AB\text{ }+\text{ }AB\text{ }+\text{ }{{B}^{2}}$

$\left[ Since,\text{ }AB\text{ }=\text{ }BA \right]$

$=\text{ }{{A}^{2~}}+\text{ }2AB\text{ }+\text{ }{{B}^{2}}$

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