If A and B are square matrices of the same order, then (A+B) (A-B) is equal to

For matrices A and B of the same order, the expansion of $(A+B)(A-B)$ requires careful attention because matrix multiplication is not commutative.

Expanding using the distributive property:

$(A+B)(A-B) = A(A-B) + B(A-B)$

Distributing A into $(A-B)$:

$A(A-B) = A^2 - AB$

Distributing B into $(A-B)$:

$B(A-B) = BA - B^2$

Combining the results:

$(A+B)(A-B) = A^2 - AB + BA - B^2$

$(A+B)(A-B) = A^2 - B^2 + BA - AB$

The expression cannot be simplified to $A^2 - B^2$ because matrix multiplication is not commutative, meaning $AB \neq BA$ in general.

Therefore, the terms $BA - AB$ do not cancel out and must remain in the final expression.

Therefore, $(A+B)(A-B) = A^2 - B^2 + BA - AB$