Question

If A and B are square matrices of the same order, explain, why in general

(A + B)² ≠ A² + 2AB + B²

Answer 1

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

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

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

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

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