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# Find the transpose of each of the following matrices : $$\text{ (i) } \begin{bmatrix} 5 \\ \tfrac{1}{2} \\ -1 \end{bmatrix} \qquad \qquad (ii) \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix} \qquad \qquad (iii)\begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}$$

Toolbox:
• If A =$[a_{ij}]$ be an matrix,then the matrix obtained by interchanging the rows and column of A is called the transpose of A.
• Transpose of the matrix A is denoted by A' or $A^T.$
$(i)\begin{bmatrix}5\\\frac{1}{2}\\-1\end{bmatrix}\Rightarrow$As per the formula interchange the rows into column to get the transpose.

$\Rightarrow \begin{bmatrix}5\\\frac{1}{2}\\-1\end{bmatrix}=A.$

$A'=\begin{bmatrix}5 & \frac{1}{2} &-1\end{bmatrix}$

$(ii)\begin{bmatrix}1 &-1\\2 & 3\end{bmatrix}$

Let A=$\begin{bmatrix}1 &-1\\2 & 3\end{bmatrix}$

$A'=\begin{bmatrix}1 &2\\-1 & 3\end{bmatrix}$

$(iii)\begin{bmatrix}-1 &5 & 6\\\sqrt 3 & 5 & 6\\2 & 3 &-1\end{bmatrix}$

Let $A=\begin{bmatrix}-1 &5 & 6\\\sqrt 3 & 5 & 6\\2 & 3 &-1\end{bmatrix}$

$A'=\begin{bmatrix}-1 &\sqrt 3 & 2\\5 & 5 & 3\\6 & 6 &-1\end{bmatrix}$