JEE Main & Advanced Mathematics Determinants & Matrices Transpose of a Matrix

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by \[{{A}^{T}}\]or \[{A}'\].

From the definition it is obvious that if order of A is \[m\times n,\] then order of \[{{A}^{T}}\]is \[n\times m\].

Example:

Transpose of matrix \[{{\left[ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}}  \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\ \end{matrix} \right]}_{2\times 3}}\] is \[\text{ }{{\left[ \begin{matrix} {{a}_{1}} & {{b}_{1}}  \\ {{a}_{2}} & {{b}_{2}}  \\ {{a}_{3}} & {{b}_{3}}  \\ \end{matrix} \right]}_{3\times 2}}\]

Properties of transpose : Let A and B be two matrices then,

(i)  \[{{({{A}^{T}})}^{T}}=A\]

(ii)  \[{{(A+B)}^{T}}={{A}^{T}}+{{B}^{T}},A\]and B being of the same order

(iii)  \[{{(kA)}^{T}}=k{{A}^{T}},k\] be any scalar (real or complex)

(iv) \[{{(AB)}^{T}}={{B}^{T}}{{A}^{T}},A\] and B being conformable for the product AB

(v) \[{{({{A}_{1}}{{A}_{2}}{{A}_{3}}.....{{A}_{n-1}}{{A}_{n}})}^{T}}={{A}_{n}}^{T}{{A}_{n-1}}^{T}.......{{A}_{3}}^{T}{{A}_{2}}^{T}{{A}_{1}}^{T}\]

(vi) \[{{I}^{T}}=I\]