Question - Matrix Multiplication and Commutativity

Solution:

Note: Commutativity does not generally hold for matrix multiplication. Therefore, \( AB \) is not necessarily equal to \( BA \). We can prove this by counterexample or logic regarding the properties of matrix multiplication.

Consider two generic \( n \times n \) matrices \( A \) and \( B \). If \( AB = BA \), then for all possible \( A \) and \( B \), the products must be equal. Since it is known that matrix multiplication is not commutative in general, there exists at least one pair of matrices \( A \) and \( B \) where \( AB \neq BA \). Therefore, even if \( BA = I \), where \( I \) is the identity matrix, it does not imply that \( AB \) also equals \( I \) or that \( AB = BA \).

Without a specific counterexample, the general claim that \( AB \) equals \( BA \) based solely on \( BA = I \) cannot be supported and is false. Thus, \( AB \) does not necessarily equal \( BA \).

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