Question

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

• A. $A^2 + B^2$
• B. $A^2 - B^2$
• C. $A + B$
• D. $A - B$

Answer 1

Answer: (B)

$A^2 - B^2$

Answer 2

$B = ABA^{-1}$ (Given) But $B = BAA^{-A}$ $\therefore \, \, ABA^{-1} = BAA^{-1} \, \, \Rightarrow \, \, AB = BA$ Now $(A + B) (A - B) = A^2 - AB + BA - B^2$ $= A^2 - AB + AB - B^2 \, \, \, [ \therefore \, \, AB = BA]$ $= A^2 - B^2$