The square of a sum for matrices, expressed as (e+f)², does not equal the sum of squares due to the non-commutative nature of matrix multiplication. The correct expansion is (e+f)(e+f) = e² + ef + fe + f², highlighting that the terms ef and fe are distinct. This distinction arises because, in matrix operations, the order of multiplication matters. The discussion emphasizes the importance of recognizing these properties when working with matrices of any size. Understanding this concept is crucial for accurate matrix calculations.