Math Problem Statement

[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

Solution

The given statement is incorrect because matrix multiplication is not commutative, meaning AB≠BAAB \neq BA in general. As a result, expanding (A+B)2(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)=A2+AB+BA+B2(A + B)^2 = (A + B)(A + B) = A^2 + AB + BA + B^2

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

A2+AB+BA+B2A^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(A + B)^3 differ?
  3. Can you expand (A−B)2(A - B)^2 in a similar manner?
  4. What happens if AA and BB commute (i.e., AB=BAAB = BA)?
  5. How would you expand (A+B+C)2(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.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Matrix Multiplication
Non-commutative Operations

Formulas

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

Theorems

Non-commutativity of Matrix Multiplication

Suitable Grade Level

Undergraduate Mathematics or Advanced High School

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