Question

Four friends Abhay, Bina, Chhaya, and Devesh were asked to simplify \( 4 AB + 3(AB + BA) - 4 BA \), where \( A \) and \( B \) are both matrices of order \( 2 \times 2 \). It is known that \( A \neq B \) and \( A^{-1} \neq B \). Their answers are given as:

• A. Abhay: \( 6 AB \)
• B. Bina: \( 7 AB - BA \)
• C. Chhaya: \( 8 AB \)
• D. Devesh: \( 7 BA - AB \)

Hint:

When simplifying matrix expressions, carefully distribute constants and combine like terms.

Answer 1

The Correct Option is B

The expression \( 4AB + 3(AB + BA) - 4BA \) requires simplification, given that \( A \) and \( B \) are \( 2 \times 2 \) matrices. The steps for simplification are as follows:

• The initial expression is:
  \( 4AB + 3(AB + BA) - 4BA \)
• Distribute the constant \( 3 \):
  \( 4AB + 3AB + 3BA - 4BA \)
• Combine terms with \( AB \) and terms with \( BA \):
  \( (4AB + 3AB) + (3BA - 4BA) = 7AB - BA \)

The simplified form of the expression is \( 7AB - BA \). This result aligns with Bina's provided answer.