====== Question 9, Exercise 2.2 ======
Solutions of Question 9 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 9=====
Consider any two particular matrices $A$ and $B$ of your choice of order $3 \times 3$ and show that $(A+B)^{t}=A^{t}+B^{t}$.

** Solution. **

Let:
\begin{align*}
A &= \begin{pmatrix} 
a_{11} & a_{12} & a_{13} \\ 
a_{21} & a_{22} & a_{23} \\ 
a_{31} & a_{32} & a_{33} 
\end{pmatrix} \\
B &= \begin{pmatrix} 
b_{11} & b_{12} & b_{13} \\ 
b_{21} & b_{22} & b_{23} \\ 
b_{31} & b_{32} & b_{33} 
\end{pmatrix}\end{align*}
\begin{align*} A + B &= 
\begin{pmatrix} 
a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13} \\ 
a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \\ 
a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33} 
\end{pmatrix}
\end{align*}
\begin{align*}
(A + B)^t &= 
\begin{pmatrix} 
a_{11} + b_{11} & a_{21} + b_{21} & a_{31} + b_{31} \\ 
a_{12} + b_{12} & a_{22} + b_{22} & a_{32} + b_{32} \\ 
a_{13} + b_{13} & a_{23} + b_{23} & a_{33} + b_{33} 
\end{pmatrix} 
\end{align*}	
\begin{align*}
A^t &= 
\begin{pmatrix} 
a_{11} & a_{21} & a_{31} \\ 
a_{12} & a_{22} & a_{32} \\ 
a_{13} & a_{23} & a_{33} 
\end{pmatrix}\\
B^t &= 
\begin{pmatrix} 
b_{11} & b_{21} & b_{31} \\ 
b_{12} & b_{22} & b_{32} \\ 
b_{13} & b_{23} & b_{33} 
\end{pmatrix} \\
A^t + B^t &=  
\begin{pmatrix} 
a_{11} + b_{11} & a_{21} + b_{21} & a_{31} + b_{31} \\ 
a_{12} + b_{12} & a_{22} + b_{22} & a_{32} + b_{32} \\ 
a_{13} + b_{13} & a_{23} + b_{23} & a_{33} + b_{33} 
\end{pmatrix} \\
\text{and}\\ (A + B)^t &=  
\begin{pmatrix} 
a_{11} + b_{11} & a_{21} + b_{21} & a_{31} + b_{31} \\ 
a_{12} + b_{12} & a_{22} + b_{22} & a_{32} + b_{32} \\ 
a_{13} + b_{13} & a_{23} + b_{23} & a_{33} + b_{33} 
\end{pmatrix} 
\end{align*}
Thus, $\quad(A + B)^t = A^t + B^t$.	


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