Transpose of a Matrix:

The matrix gets from any provided matrix A, by interchanging columns and rows, is known as the transpose of A and is shows by A'.

If A = [a_ij]_mxn­ and A' = [b­_ij]_nxm then b_ij = a_ji, ∀ i, j

 For example:  If   A = , then 

Properties of Transposes:

(i).        (A')' = A.

(ii).       (A ± B)' = A' ± B', A and B being conformable matrices.

(iii).      (αA)' = αA', a being scalar.

(iv).      (AB)' = B'A', A and B being conformable for multiplication.

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