JEE MAIN - Mathematics (2006 - No. 7)
If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that
$${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then which of the following will be always true?
$${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then which of the following will be always true?
$$A=B$$
$$AB=BA$$
either of $$A$$ or $$B$$ is a zero matrix
either of $$A$$ or $$B$$ is identity matrix
Explication
$${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right)$$
$${A^2} - {B^2} = {A^2} + AB - BA - {B^2}$$
$$ \Rightarrow AB = BA$$
$${A^2} - {B^2} = {A^2} + AB - BA - {B^2}$$
$$ \Rightarrow AB = BA$$