Q.If $A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$, find the transpose $A^T$, and verify that $(A^T)^T=A$.

The transpose is formed by turning each row of the original matrix into a column; taking the transpose a second time always returns the original matrix.

Forming the transpose

$A$ is of order $2\times3$, so $A^T$ is of order $3\times2$. Row 1 of $A$, $(1,2,3)$, becomes column 1 of $A^T$; row 2, $(4,5,6)$, becomes column 2:

Verifying $(A^T)^T=A$

Transposing $A^T$ (order $3\times2$) turns its rows back into columns, giving a matrix of order $2\times3$: row 1 of $A^T$, $(1,4)$, becomes column 1; row 2, $(2,5)$, becomes column 2; row 3, $(3,6)$, becomes column 3:

$(A^T)^T=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}=A$ …

Forgetting that the order of the matrix changes when transposed (a $2\times3$ matrix becomes $3\times2$), and then trying to compare $A$ an …