Transpose of a Matrix

If the rows and columns of a matrix are interchanged, i.e. the first row becomes the first column, the second row becomes the second column, the third row becomes the third column, e.t.c, then the new matrix so formed is called the transpose of the original matrix.

If A is the original matrix, its transpose is denoted by \(\scriptsize \tilde{A} \;or\; A^T\).

Examples: 

If A = \( \scriptsize \begin{pmatrix} 4 & 6 \\ 7 & 9 \\ 2 & 5 \end{pmatrix}\)

Then \(\scriptsize \tilde{A} = \begin{pmatrix} 4 & 7 & 2 \\ 6 & 9 & 5 \end{pmatrix}\)

B = \( \scriptsize \begin{pmatrix} 2 & 7 & 6 \\ 6 & 1 & 5 \end{pmatrix}\)

Then \(\scriptsize \tilde{B} = \begin{pmatrix} 2 & 6 \\ 7 & 1 \\ 6 & 5 \end{pmatrix}\)