Transpose of a matrix product is the reversed product of the two transposes

The transpose of a matrix product is equal to the reversed product of the two transposes . To prove this, let us first recall that an element in is given as

and the transpose of a matrix is given as

which implies that

So this means that

or equivalently,

Now let us consider . Substituting into the definition of the matrix product Meanwhile, from the definition of the matrix product,

Since scalar multiplication is commutative, this is identical to the definition of . Hence .