The transpose of an m×n matrix is the n×m matrix obtained by interchanging the rows and columns. The transpose of A is denoted by A^T, A^t, or A′. Thus, if A = [a_ij], then $A^T=\left[a_{\mathrm{ij}}\prime \right]$, where $a_{\mathrm{ij}}\prime =a_{ji}$; that is,

The following properties hold, for matrices A and B of appropriate orders:

1. (i) (A^T)^T = A.

2. (ii) (A + B)^T = A^T + B^T.

3. (iii) (kA)^T = kA^T.

4. (iv) (AB)^T = B^TA^T.