Question

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 correct?
A. \[AB = BA\]
B. Either \[A\] or \[B\] is a zero matrix
C. Either \[A\] or \[B\] is an identity matrix
D. \[A = B\]

Answer 1

Hint: Given that \[A\] and \[B\] are square matrices of size \[n \times n\] and an expression \[{A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right)\] is given. You have to check which option is correct among the given four options. Check each option one by one whether it is true or not.

Formula used
Distributive law: \[A\left( {B + C} \right) = AB + AC\], where \[A,B,C\] are three matrices.
In matrix, \[AB \ne BA\], in general.

Complete step by step solution
Given that \[{A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right)\], where \[A\]and \[B\] are square matrices of size \[n \times n\]
Now, \[\left( {A - B} \right)\left( {A + B} \right)\]
\[ = A\left( {A + B} \right) - B\left( {A + B} \right)\], using distributive law
\[ = {A^2} + AB - BA - {B^2}\]
In matrix, \[AB \ne BA\], in general.
\[\therefore {A^2} - {B^2} = {A^2} + AB - BA - {B^2}\]
Cancel \[{A^2}\] and \[{B^2}\] from both sides.
\[ \Rightarrow AB - BA = O\], where \[O\] is a null matrix of size \[n \times n\]
\[ \Rightarrow AB = BA\]

Hence option A is correct.

Note:Remember that in matrix, \[AB \ne BA\], in general. Adding or subtracting the same matrices, the null matrix is obtained. Don’t write \[0\] in the place of \[O\]. If \[AB = BA\], it is not necessary that either \[A\] or \[B\] is a zero matrix or identity matrix or they are equal.