x AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:(AT)i,j = Aj,i.[1230−67]T=[102−637]{\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{T}={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e., the matrix sum does not depend on the order of the summands: A + B = B + A.[4]The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A.Row operations are ways to change matrices. There are three types of row operations: row switching, that is interchanging two rows of a matrix; row multiplication, multiplying all entries of a row by a non-zero constant; and finally row addition, which means adding a multiple of a row to another row. These row operations are used in a number of ways in