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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