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 $A^{\top}$. Hence $A^{\prime}=\left[a_{j i}\right]_{n \times m}$ 

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) $(k A)^{\prime}=k A^{\prime}$
iv) $(A B)^{\prime}=B^{\prime} A^{\prime}($ given that $A$ and $B$ are conformable for matrix product $A B$ )