Question
Let \(A\) and \(B\) be square matrices of the same order. Does \[ (A + B)^2 = A^2 + 2AB + B^2 \] hold? If not, why?
Solution
Step 1: Expand \((A+B)^2\)
\[ (A + B)^2 = (A + B)(A + B) \] \[ = A^2 + AB + BA + B^2 \]Step 2: Compare with Algebra
In algebra: \[ (a+b)^2 = a^2 + 2ab + b^2 \] But for matrices: \[ AB \ne BA \quad \text{(in general)} \]Step 3: Conclusion
\[ (A+B)^2 = A^2 + AB + BA + B^2 \] This equals \(A^2 + 2AB + B^2\) only if: \[ AB = BA \]Final Answer
\[
(A+B)^2 \ne A^2 + 2AB + B^2 \quad \text{(in general)}
\]
\[
\text{Because } AB \ne BA
\]
\[
\text{Equality holds only when } AB = BA
\]