(AB)^T = A^T * B^T
The statement “(AB)^T = A^T * B^T” is true.
The statement “(AB)^T = A^T * B^T” is true.
The transpose of a matrix is an operation that flips the matrix over its main diagonal, reflecting its elements across the diagonal. When taking the transpose of a product of matrices, the order of the matrices is reversed.
To prove the statement, let’s consider two matrices A and B:
(AB)^T is the transpose of the product AB, which means we flip the product over its main diagonal.
A^T * B^T is the product of the transposes of A and B, which means we first transpose A, then transpose B, and finally multiply them.
By the properties of matrix multiplication and transposition, it can be shown that the transpose of a product is equal to the product of the transposes, given that the matrices involved have compatible dimensions. Therefore, the statement is true:
(AB)^T = B^T * A^T = A^T * B^T
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