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#### Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 1 Subquestion (i).

Given:

$A=\begin{bmatrix} 2 &-3 \\ -7 &5 \end{bmatrix}, B=\begin{bmatrix} 1 &0 \\ 2 & -4 \end{bmatrix}$

To prove:   $\left ( 2A \right )^{T}=2A^{T}$

Hint: The $A^{T}$ of matrix $A$ can be obtained by reflecting the elements along its main diagonal

Solution:
$\! \! \! \! \! \! \! \! A=\begin{bmatrix} 2 &-3 \\ -7 & 5 \end{bmatrix}, A^{T}=\begin{bmatrix} 2 &-7 \\ - 3& 5 \end{bmatrix}\\\\ 2A=2\begin{bmatrix} 2 &-3 \\ -7 & 5 \end{bmatrix}\\\\ 2A=\begin{bmatrix} 4 & -6\\ -14 & 10 \end{bmatrix}\\\\$

$\left ( 2A \right )^{T}=\begin{bmatrix} 4 &-14 \\ -6& 10 \end{bmatrix}$                                                                                                                      …… (1)

$2A ^{T}=2\begin{bmatrix} 2 & -7\\ -3 & 5 \end{bmatrix}=\begin{bmatrix} 4 &-14 \\ -6& 10 \end{bmatrix}$                                                                                              …… (2)

From 1 & 2

$\left ( 2A \right ) ^{T}=2A^{T}$