## The transpose of a product of matrices equals the product of their transposes in the same order.

### False because it should be reverse order not same order. (AB)^T = B^T * A^T

The statement The transpose of a product of matrices equals the product of their transposes in the same order is not true in general.

Specifically, if A and B are matrices of compatible sizes (meaning the number of columns of A equals the number of rows of B), then we have:

(AB)^T = B^T A^T

This is known as the transpose of a product rule, and it states that the transpose of a product of matrices equals the product of their transposes taken in the opposite order.

To see why this is true, consider the (i,j)th entry of the product AB:

(AB){i,j} = \sum{k=1}^n a_{i,k} b_{k,j}

Then, the (j,i)th entry of the product B^T A^T is:

(B^T A^T){j,i} = \sum{k=1}^n b_{j,k} a_{k,i}

Comparing the two sums, we see that they are equal (up to the order of summation), so the (j,i)th entry of B^T A^T equals the (i,j)th entry of AB. Therefore, we have:

(AB)^T = B^T A^T

So, in general, the transpose of a product of matrices does not equal the product of their transposes in the same order, but rather in the opposite order.

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