[Maximum mark: 6]
Let A, B and C be non-singular 2x2 matrices, / the 2x2 identity matrix and k a scalar.
The following statements are incorrect. For each statement, write down the correct version of the right hand side.

(A + B)^2= A^2 + 2AB + B^2

The given statement is incorrect because matrix multiplication is not commutative, meaning \( AB \neq BA \) in general. As a result, expanding \( (A + B)^2 \) requires considering the order of matrix multiplication. Here's how it should be expanded correctly:

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

Thus, the correct version of the right-hand side is:

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

Would you like further details or clarification? 

Here are five related questions to deepen your understanding:
1. What is matrix multiplication, and why is it non-commutative?
2. How would the expansion of \( (A + B)^3 \) differ?
3. Can you expand \( (A - B)^2 \) in a similar manner?
4. What happens if \( A \) and \( B \) commute (i.e., \( AB = BA \))?
5. How would you expand \( (A + B + C)^2 \) for three matrices?

**Tip:** Always pay attention to the order of multiplication when working with matrices, as it can affect the outcome significantly.

Let A, B and C be non-singular 2x2 matrices, I the 2x2 identity matrix and k a scalar. The following statements are incorrect. For each statement, write down the correct version of the right-hand side: (A + B)^2 = A^2 + 2AB + B^2.

Learn how to correctly expand (A + B)^2 when working with non-singular 2x2 matrices. This problem explores the non-commutative nature of matrix multiplication, and provides step-by-step reasoning for obtaining the correct expansion: A^2 + AB + BA + B^2. Suitable for advanced high school students and undergraduates.

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