Transpose of a Matrix - Practice Questions & MCQ

Transpose of a matrix

In simple language, the transpose of a matrix is changing its rows into columns or columns into rows. Let $\mathrm{A}={\left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{\mathrm{m}\times \mathrm{n}}$ be a matrix, then matrix obtained by changing rows into columns or vice-versa will give transpose of $A$ which is denoted as $A^{\prime }$ or $A^{\mathrm{\top }}$. Hence $A^{\prime }={\left[a_{ji}\right]}_{n\times m}$ 

E.g 

$\mathrm{A}=\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\Rightarrow {\mathrm{A}}^{\prime }=\left[\begin{array}{lll}{a}_{11}& a_{21}& a_{31}\\ a_{12}& a_{22}& a_{32}\\ a_{13}& a_{23}& a_{33}\end{array}\right]$
If, $A=\left[\begin{array}{ll}2& 6\\ 3& 7\\ 5& 8\end{array}\right]$ then, $A^{\prime }=\left[\begin{array}{lll}2& 3& 5\\ 6& 7& 8\end{array}\right]$

Properties of the transpose of a matrix:

If $A^{\prime }$ and $B^{\prime }$ denote the transpose of the matrices $A$ and $B$, then :
i) ${\left(A^{\prime }\right)}^{\prime }=A$
ii) $(A\pm B{)}^{\prime }=A^{\prime }\pm B^{\prime }$ (given that $A$ and $B$ are conformable for matrix addition)
iii) $(kA{)}^{\prime }=k{A}^{\prime }$
iv) $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }($ given that $A$ and $B$ are conformable for matrix product $AB$ )