# . Find At, Bt and Verify that , (A B)T = Bt + at - Mathematics

Sum

A = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1\end{bmatrix} and B = \begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix} . Find ATBT and verify that ,

(A B)T = BT + AT

#### Solution

$AB = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1\end{bmatrix} \begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}$
$\Rightarrow AB = \begin{bmatrix}1 - 2 + 0 & 2 - 1 + 0 & 3 - 3 + 0 \\ 2 + 2 + 0 & 4 + 1 + 3 & 6 + 3 + 3 \\ 1 + 4 + 0 & 2 + 2 + 1 & 3 + 6 + 1\end{bmatrix}$
$\Rightarrow AB = \begin{bmatrix}- 1 & 1 & 0 \\ 4 & 8 & 12 \\ 5 & 5 & 10\end{bmatrix}$
$\Rightarrow \left( AB \right)^T = \begin{bmatrix}- 1 & 4 & 5 \\ 1 & 8 & 5 \\ 0 & 12 & 10\end{bmatrix} . . . \left( 1 \right)$
$Now,$
$B^T A^T = \begin{bmatrix}1 & 2 & 0 \\ 2 & 1 & 1 \\ 3 & 3 & 1\end{bmatrix}\begin{bmatrix}1 & 2 & 1 \\ - 1 & 1 & 2 \\ 0 & 3 & 1\end{bmatrix}$
$\Rightarrow B^T A^T = \begin{bmatrix}1 - 2 + 0 & 2 + 2 + 0 & 1 + 4 + 0 \\ 2 - 1 + 0 & 4 + 1 + 3 & 2 + 2 + 1 \\ 3 - 3 + 0 & 6 + 3 + 3 & 3 + 6 + 1\end{bmatrix}$
$\Rightarrow B^T A^T = \begin{bmatrix}- 1 & 4 & 5 \\ 1 & 8 & 5 \\ 0 & 12 & 10\end{bmatrix} . . . \left( 2 \right)$
$\Rightarrow \left( AB \right)^T = B^T A^T \left[ {\text{From eqs}} . (1) \hspace{0.167em} and (2) \right]$

Concept: Multiplication of Two Matrices
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 5 Algebra of Matrices
Exercise 5.4 | Q 3.2 | Page 54