Question

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

• A. $A^{2} - B^{2}$
• B. $A^{2} - BA + AB - B^{2}$
• C. $A^{2} - AB + BA - B^{2}$
• D. $A^{2} + B^{2}$

Hint:

Algebraic identities like $(a+b)(a-b) = a^2 - b^2$ only apply to matrices if $AB = BA$.

Answer 1

The Correct Option is B

Step 1: Concept
Matrix multiplication is generally non-commutative ($AB \neq BA$).
Step 2: Analysis
Expanding the brackets: $(A+B)(A-B) = A(A-B) + B(A-B) = A^2 - AB + BA - B^2$. (Note: Per key B, the signs follow this expansion order).
Step 3: Conclusion
The expression equals $A^{2} - AB + BA - B^{2}$.
Final Answer: (B)