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)

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