#### Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 2

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

Given:$A=\begin{bmatrix} 3\\ 5\\ 1 \end{bmatrix}, B=\begin{bmatrix} 1 &0 &4 \end{bmatrix}$

Hint: The $A^{T}$ of matrix$A$ can be obtained by reflecting the elements along it’s main diagonal.

Solution:

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

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

$\left ( AB \right )^{T}=\begin{bmatrix} 3 &5 &5 \\ 0& 0 & 0\\ 12 & 20 & 8 \end{bmatrix}$                                                                                      … (1)

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

$=\begin{bmatrix} 3 &5 &2 \\ 0& 0 & 0\\ 12 & 20 & 8 \end{bmatrix}$                                                                                              ….. (2)

(1) &( 2)

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