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

Answer: $\left ( A+B\right )^{T}=A^{T}+B^{T}$

Given:

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

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

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

Solution:

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

R.H.S:

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

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

$=\begin{bmatrix} 3 & -5\\ -3& -1 \end{bmatrix}$                                                                                         …… (1)

$\left ( A+B \right )^{T}=\left ( \begin{bmatrix} 2 & -3\\ -7 & 5 \end{bmatrix}+\begin{bmatrix} 1 &0\\ 2 &-4 \end{bmatrix} \right )$

$=\left ( \begin{bmatrix} 2+1 & -3+0\\ -7+2 & 5-4 \end{bmatrix} \right )$

$=\left ( \begin{bmatrix} 3 & -3\\ -5 & 1 \end{bmatrix} \right )^{T}$

$= \begin{bmatrix} 3 & -5\\ -3 & 1 \end{bmatrix} -\left ( 2 \right )$

From 1 &  2

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