Theorem. The following properties hold:

*(A*, that is the transpose of the transpose of^{T})^{T}=A*A*is*A*(the operation of taking the transpose is an involution).-
*(A+B)*, the transpose of a sum is the sum of transposes.^{T}=A^{T}+B^{T} -
*(kA)*.^{T}=kA^{T} -
*(AB)*, the transpose of a product is the product of the transposes in the reverse order.^{T}=B^{T}A^{T}

** Proof.** 1. The (*i,j*)-entry of *A ^{T}* is the (

2. The (*i,j*)-entry of *A ^{T}+B^{T}* is the sum of (

3. The proof is similar.

4. Compare the (*i,j*)-entries of *(AB) ^{T}* and