Q.Find the transpose of $A=\begin{pmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{pmatrix}$, and state its order.

The transpose is formed by turning each row of $A$ into the corresponding column.

To transpose, write the first row of $A$, namely $(2,-1,5)$, as the first column, and the second row $(0,3,4)$ as the second column: $A^{\mathsf{T}}=\begin{pmatrix} 2 & 0 \\ -1 & 3 \\ 5 & 4 \end{pmatrix}.$

Order. $A$ is of order $2\times 3$, so $A^{\mathsf{T}}$ is of order $3\times 2$ (rows and columns swap). …

A common slip is to keep the order 2 by 3 for the transpose. Transposing swaps the order, so …