To determine which of the given options is true for any two square matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we need to carefully review each statement and understand the properties of matrix operations. Let's evaluate each option one by one:
a) [tex]\((A B)^T = A^{\top} B^T\)[/tex]
The transpose of a product of two matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is not equal to [tex]\(A^{\top} B^T\)[/tex]. Instead, it follows a specific rule. Hence, this statement is false.
b) [tex]\((A B)^T = B^T A^T\)[/tex]
This is indeed a property of matrix transposes. When taking the transpose of a product of two matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], the transpose of the product is equal to the product of the transposes in reverse order. Thus, [tex]\((A B)^T\)[/tex] is equal to [tex]\(B^T A^T\)[/tex]. Hence, this statement is true.
c) [tex]\(A B = B A\)[/tex]
This statement would mean that matrix multiplication is commutative. However, in general, matrix multiplication is not commutative. There are specific cases where it can be, such as [tex]\(A\)[/tex] and [tex]\(B\)[/tex] being special types of matrices (e.g., diagonal matrices), but in general, [tex]\(A B\)[/tex] is not equal to [tex]\(B A\)[/tex]. Hence, this statement is false.
d) [tex]\(A - B = B - A\)[/tex]
This simplification suggests that the order of subtraction does not matter, which would imply that [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are the same matrices because [tex]\(A - B = -(B - A)\)[/tex]. In general, for any two matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], this is not true unless [tex]\(A = B\)[/tex]. Hence, this statement is false.
Given these evaluations, the true statement is:
b) [tex]\((A B)^T = B^T A^T\)[/tex]
Thus, the correct answer is option b.
