Question

Let A and B be square matrices of the same order. Does (A + B)² = A² + 2AB + B² hold? If not, why?

Answer 1

$$\begin{gathered} \mathrm{LHS}={\left(A+B\right)}^2 \\ =\left(A+B\right)\left(A+B\right) \\ =A\left(A+B\right)+B\left(A+B\right) \\ =A^2+AB+BA+B^2 \end{gathered}$$

We know that a matrix does not have commutative property. So,
AB ≠ BA
Thus,
${\left(A+B\right)}^2$ ≠ $A^2+2AB+B^2$