3.2 Transpose

What is transpose

• the transpose of mxn matrix A is the nxm matrix, denoted $A^T,$ whose $(i,j)$-entry is the same as the $j,i$-entry of A

$$(A^T{)}_{ij}=(A{)}_{ji}⟹A=\left[\begin{array}{ccc}1& 0& 1\\ 2& 3& 4\end{array}\right]⟹A^T=\left[\begin{array}{cc}1& 2\\ 0& 3\\ 1& 4\end{array}\right]$$

• Properties of transpose

  • $(A^T{)}^T=A$
  • $(A+B{)}^T=A^T+B^T$
  • $(AB{)}^T=B^T{A}^T$
  • $(sA{)}^T=S(A^T)$