Question

If \( A \) and \( B \) are matrices and \( AB = BA = A^{-1} \) then the value of \( (A + B)(A - B) \) is

• A. \( A^2 + B^2 \)
• B. \( A^2 - B^2 \)
• C. \( A + B \)
• D. \( A - B \)

Hint:

The identity \( (A + B)(A - B) = A^2 - B^2 \) holds true for matrices, just like it does for real numbers.

Answer 1

The Correct Option is B

Step 1: Matrix Multiplication.
Using the distributive property of matrices and the fact that \( AB = BA = A^{-1} \), we find: \[ (A + B)(A - B) = A^2 - B^2 \] Thus, the expression simplifies to \( A^2 - B^2 \).
Step 2: Conclusion.
The correct answer is (B), \( A^2 - B^2 \).