Properties of the Transpose

Suppose we have two matrices $A$ and $B$ of the same dimension (so we can add them).

Then $(A + B)^T = A^T + B^T$.

To see this, let’s consider an example with the following matrices:

$$A = \begin{bmatrix} 1 & 0 & 4 \\ -2 & -3 & -6 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & -2 & -4 \\ 0 & 1 & -1 \end{bmatrix}.$$

Then:

$$\begin{align*} (A + B)^T &= \left( \begin{bmatrix} 1 & 0 & 4 \\ -2 & -3 & -6 \end{bmatrix} + \begin{bmatrix} 6 & -2 & -4 \\ 0 & 1 & -1 \end{bmatrix} \right)^T \\ &= \begin{bmatrix} 7 & -2 & 0 \\ -2 & -2 & -7 \end{bmatrix}^T \\ &= \begin{bmatrix} 7 & -2 \\ -2 & -2 \\ 0 & -7 \end{bmatrix}. \end{align*}$$

Note that:

$$A^T = \begin{bmatrix} 1 & -2 \\ 0 & -3 \\ 4 & -6 \end{bmatrix}, \quad B^T = \begin{bmatrix} 6 & 0 \\ -2 & 1 \\ -4 & -1 \end{bmatrix}.$$

So:

\begin{align*} A^T + B^T &= \begin{bmatrix} 1 & -2 \\ 0 & -3 \\ 4 & -6 \end{bmatrix} + \begin{bmatrix} 6 & 0 \\ -2 & 1 \\ -4 & -1 \end{bma ...