Math Problem Statement
Let A and B be n x n matrices. Prove that (A + B)^2 = A^2 + 2AB + B^2 if and only if AB = BA by filling in the blank.
Solution
This problem asks you to prove that if and only if , by filling in the appropriate steps in the blank.
Step-by-Step Analysis:
Step 1: Expand
= A^2 + AB + BA + B^2$$ #### Step 2: Compare the expanded result to the desired equality The problem states that $$(A + B)^2 = A^2 + 2AB + B^2$$. For this equality to hold: $$A^2 + AB + BA + B^2 = A^2 + 2AB + B^2$$ #### Step 3: Analyze the condition By subtracting $$A^2 + B^2$$ from both sides: $$AB + BA = 2AB$$ Simplify this: $$BA = AB$$ Thus, the condition $$AB = BA$$ is necessary for the equality to hold. --- #### Step 4: Logical Flow of Proof (Filling the Blanks): 1. Start with the expansion $$(A + B)^2 = A^2 + AB + BA + B^2$$. 2. This simplifies to $$A^2 + AB + BA + B^2 \iff A^2 + 2AB + B^2$$. 3. The condition $$AB = BA$$ ensures the equality. 4. Conclude that $$(A + B)^2 = A^2 + 2AB + B^2 \iff AB = BA$$. --- Let me know if you would like detailed feedback or further clarification! --- ### 5 Related Questions: 1. What properties do matrices need to satisfy for $$AB = BA$$? 2. How does matrix commutativity affect higher powers, such as $$(A + B)^3$$? 3. Why does the non-commutativity of matrices complicate expansions like $$(A + B)^2$$? 4. Can you provide examples of commutative and non-commutative matrices? 5. How does associativity differ from commutativity in matrix operations? --- ### Tip: When expanding matrix expressions, always check whether matrix multiplication is commutative ($$AB = BA$$) before simplifying terms!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Commutativity in Matrices
Formulas
(A + B)^2 = A^2 + AB + BA + B^2
(A + B)^2 = A^2 + 2AB + B^2
Theorems
Matrix Commutativity (AB = BA)
Suitable Grade Level
Undergraduate
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