# JEE Main & Advanced Mathematics Matrices Transpose of a Matrix

## Transpose of a Matrix

Category : JEE Main & Advanced

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) ${{I}^{T}}=I$

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