Skill: Apply properties of matrix transposes

Transposing a matrix means switching its rows and columns. An $m\times n$ matrix becomes an $n\times m$ matrix. For real matrices, useful rules are

$$(A+B)^T=A^T+B^T,\qquad (kA)^T=kA^T,\qquad (AB)^T=B^TA^T,\qquad (A^T)^T=A.$$

The key detail in a product is that the order reverses.

Example: Let

$$A=\begin{pmatrix}1&2\\0&-1\end{pmatrix},\quad B=\begin{pmatrix}3&1\\2&0\end{pmatrix},\quad C=\begin{pmatrix}1&0\\2&1\end{pmatrix}.$$

Find $\bigl((2A+B)C\bigr)^T$.

First, apply the product rule, reversing the order of the factors:

$$\bigl((2A+B)C\bigr)^T=C^T(2A+B)^T.$$

Next, use the scalar and addition rules:

$$C^T(2A+B)^T=C^T(2A^T+B^T).$$

To check the result, calculate inside the original expression:

$$2A+B=\begin{pmatrix}5&5\\2&-2\end{pmatrix}, \qquad (2A+B)C=\begin{pmatrix}15&5\\-2&-2\end{pmatrix}.$$

Therefore,

$$\bigl((2A+B)C\bigr)^T =\begin{pmatrix}15&-2\\5&-2\end{pmatrix}.$$

When transposing a product, transpose each factor and write the factors in reverse order.