Question

If $A$ and $B$ are two matrices and $\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}$ , then which of the following options is correct?
A. $AB = BA$
B. ${A^2} + {B^2} = {A^2} - {B^2}$
C. $A'B' = AB$
D. None of these

Answer 1

Hint:While opening the parentheses, and performing matrix multiplication on the left-hand side of the given equation, keep in mind that matrix multiplication is not commutative. By saying matrix multiplication is not commutative, we mean that on considering two matrices $A$ and $B$, $AB \ne BE$.

Complete step by step Solution:
We are given two matrices, $A$ and $B$ such that $\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}$ .
Performing Matrix Multiplication of the left-hand side,
$\left( {A + B} \right)\left( {A - B} \right) = AA - AB + BA - BB$
Now, $XX = {X^2}$ , therefore
$\left( {A + B} \right)\left( {A - B} \right) = {A^2} - AB + BA - {B^2}$
As, Left-hand side of the equation is equal to the right-hand side, hence,${A^2} - AB + BA - {B^2} = {A^2} - {B^2}$
On further simplification,
$ - AB + BA = 0$
As we know that matrix multiplication is not commutative, therefore, these two terms will not cancel out each other.
This gives: $AB = BA$
Hence, if $\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}$ , then $AB = BA$ .

Therefore, the correct option is (A).

Note: The commutative property does not hold for matrix multiplication, that is, $AB \ne BA$ . It also does not hold for matrix subtraction, that is, $A - B \ne B - A$. However, it does hold for matrix addition, that is, $A + B = B + A$ .