If A and B are 2 × 2 matrices, then which of the following is true?

The correct option is: (A – B)(A + B) = A² + AB – BA – B².

To understand why this statement is true, we must look at the properties of matrix multiplication. Unlike multiplication of real numbers, matrix multiplication is non-commutative. This means that for any two matrices $A$ and $B$, the product $AB$ is generally not equal to $BA$:
$AB\ne BA$

Because order matters in matrix multiplication, we must be careful when expanding algebraic expressions. We apply the distributive property of matrix multiplication over addition and subtraction:

For the expression $(A\−B)(A+B)$, we distribute each term:
$(A\−B)(A+B)=A(A+B)\−B(A+B)$

Now, we expand further by multiplying from the left:
$=AA+AB\−BA\−BB$

Since multiplying a matrix by itself is written as a power (i.e., $AA=A^2$ and $BB=B^2$), we substitute these in:
$=A^2+AB\−BA\−B^2$

Since $AB$ and $BA$ do not necessarily cancel out, this expression cannot be simplified any further. Thus, the identity is:
$(A\−B)(A+B)=A^2+AB\−BA\−B^2$