====== Question 8, Exercise 2.2 ======
Solutions of Question 8 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. 

=====Question 8=====
 Consider any two particular matrices $A$ and $B$ of your choice of order $2 \times 3$ and $3 \times 2$ respectively and show that $(A B)^{t}=B^{t} A^{t}$.


** Solution. **

Let's consider matrices \( A \) and \( B \) of orders \( 2 \times 3 \) and \( 3 \times 2 \).
	
Let
\begin{align*}
	A &= \begin{bmatrix}
	a_{11} & a_{12} & a_{13} \\
	a_{21} & a_{22} & a_{23}
\end{bmatrix}\\
B &= \begin{bmatrix}
	b_{11} & b_{12} \\
	b_{21} & b_{22} \\
	b_{31} & b_{32}
\end{bmatrix}\\
AB &= \begin{bmatrix}
	a_{11} & a_{12} & a_{13} \\
	a_{21} & a_{22} & a_{23}
\end{bmatrix}
\begin{bmatrix}
	b_{11} & b_{12} \\
	b_{21} & b_{22} \\
	b_{31} & b_{32}
\end{bmatrix}\\
\implies AB &= \begin{bmatrix}
	a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} \\
	a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32}
\end{bmatrix}\\
(AB)^t &= \begin{bmatrix}
	a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} \\
	a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32}
\end{bmatrix} \end{align*}
\begin{align*}A^t &= \begin{bmatrix}
	a_{11} & a_{21} \\
	a_{12} & a_{22} \\
	a_{13} & a_{23}
\end{bmatrix}\\
B^t &= \begin{bmatrix}
	b_{11} & b_{21} & b_{31} \\
	b_{12} & b_{22} & b_{32}
\end{bmatrix}\\
B^t A^t &= \begin{bmatrix}
	b_{11} & b_{21} & b_{31} \\
	b_{12} & b_{22} & b_{32}
\end{bmatrix}
\begin{bmatrix}
	a_{11} & a_{21} \\
	a_{12} & a_{22} \\
	a_{13} & a_{23}
\end{bmatrix}\\
&=\begin{bmatrix}
	a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} \\
	a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32}
\end{bmatrix}
\end{align*}

Hence $$(A B)^{t}=B^{t} A^{t}$$
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