Skill: Determine the transpose of a matrix

To find the transpose of a matrix, interchange its rows and columns. Each entry in row $i$, column $j$ moves to row $j$, column $i$. The matrix’s dimensions switch from $m \times n$ to $n \times m$.

For example, find the transpose of

$$A=\begin{bmatrix} 1 & 4 & 7\\ 2 & 5 & 8 \end{bmatrix}.$$

1. The first row, $[1\ \ 4\ \ 7]$, becomes the first column.
2. The second row, $[2\ \ 5\ \ 8]$, becomes the second column.

Therefore,

$$A^T=\begin{bmatrix} 1 & 2\\ 4 & 5\\ 7 & 8 \end{bmatrix}.$$

The original matrix is $2 \times 3$, and its transpose is $3 \times 2$. All the entries stay the same; only their positions change.