The transpose of an $$m \times n$$ matrix $$A$$, denoted by $$A^\mathsf{T}$$, is the $$n \times m$$ matrix such that the $$(j,i)$$-entry is given by $$A_{i,j}$$ for $$i = 1,\ldots,m$$ and $$j= 1,\ldots, n$$. In other words, column $$i$$ of $$A^\mathsf{T}$$ comes from row $$i$$ of $$A$$.

### Example 1

Let $$A=\begin{bmatrix} 1 & 2 & 3\\4 & 5 &6\end{bmatrix}$$. Then $$A^\mathsf{T} = \begin{bmatrix} 1 & 4 \\ 2 & 5\\ 3 & 6\end{bmatrix}$$.

### Example 2

Let $$A = \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix}$$. Then $$A^\mathsf{T}=\begin{bmatrix} 7 & 8 & 9\end{bmatrix}$$.

The following properties hold:

• $$(A^\mathsf{T})^\mathsf{T} = A$$

• If $$\lambda$$ is a scalar, then $$(\lambda A)^\mathsf{T} = \lambda A^\mathsf{T}$$

• $$(A^{-1})^\mathsf{T} = (A^\mathsf{T})^{-1}$$

• $$(A+B)^\mathsf{T} = A^\mathsf{T}+B^\mathsf{T}$$ for any $$B$$ having the same dimension as $$A$$

• If $$A$$ is $$m\times p$$ and $$B$$ is $$p \times n$$, then $$(AB)^\mathsf{T} = B^\mathsf{T}A^\mathsf{T}$$.

## Exercises

1. Express each of the following in the simplest possible form.
1. $$\begin{bmatrix} 7 & 2 \end{bmatrix} -\begin{bmatrix} 3 \\ 4 \end{bmatrix}^\mathsf{T}$$

2. $$\left(\begin{bmatrix} 1 & 2 \\ 5 & 6 \end{bmatrix}^\mathsf{T} + 2\begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix}\right)^\mathsf{T}$$