<p>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.</p> <p>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 \( AB = BA \).</p> <p>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 \( BA \).</p>
A square matrix is a matrix that has the same number of rows and columns. We can determine if a matrix is square by checking their dimensions. Let's look at the matrices one by one: Matrix A: \[\begin{bmatrix} -1 & 4 & -2 \\ -4 & 3 & 5 \\ -3 & 1 & 6 \end{bmatrix}\] Matrix A has 3 rows and 3 columns, so it is a 3x3 matrix. Matrix B: \[\begin{bmatrix} 5 & 2 & -1 \\ 1 & 3 & 4 \\ \end{bmatrix}\] Matrix B has 2 rows and 3 columns, so it is a 2x3 matrix and not square. Matrix C: \[\begin{bmatrix} 12 & 6 \\ 5 & 3 \\ 0 & 7 \\ \end{bmatrix}\] Matrix C has 3 rows and 2 columns, so it is a 3x2 matrix and not square. Only Matrix A is a square matrix because it has the same number of rows and columns (3x3).
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