The transpose of a matrix is obtained by interchanging rows and columns. In terms of components, $$\left(A_{ij}\right)^T=A_{ji}.$$ For example, $$A = \left(\begin{array}{cc} a&b\\ c&d\\ \end{array} \right) \Longrightarrow A^T= \left(\begin{array}{cc} a&c\\ b&d\\ \end{array} \right)$$ and $$B = \left(\begin{array}{ccc} a&b&c\\ d&e&f\\ g&h&i\\ \end{array} \right) \Longrightarrow B^T= \left(\begin{array}{ccc} a&d&g\\ b&e&h\\ c&f&i\\ \end{array} \right).$$

A square matrix is called symmetric if it is equal to its transpose, that is, if $A = A^T$.

Non-square matrices also have transposes, for example $$v = \left(\begin{array}{c} x\\ y\\ z\\ \end{array} \right) \Longrightarrow v^T= \left(\begin{array}{ccc} x&y&z\\ \end{array} \right).$$