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

\[LHS = \left( A + B \right)^2 \]

\[ = \left( A + B \right)\left( A + B \right)\]

\[ = A\left( A + B \right) + B\left( A + B \right)\]

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

We know that a matrix does not have commutative property. So,
AB ≠ BA
Thus,

\[\left( A + B \right)^2\]≠ 

\[A^2 + 2AB + B^2\]