Answer:
Let matrix [tex]A = \left[\begin{array}{ccc}1&2\\3&4\\\end{array}\right][/tex]
Let matrix [tex]B = \left[\begin{array}{ccc}5&6\\7&8\\\end{array}\right][/tex]
Matrix [tex]A^{T} = \left[\begin{array}{ccc}1&3\\2&4\\\end{array}\right][/tex]
Matrix [tex]B^{T} = \left[\begin{array}{ccc}5&7\\6&8\\\end{array}\right][/tex]
1. Solving for [tex](A + B)^{T}[/tex]
Firstly determine (A + B), then solve [tex](A + B)^{T}[/tex]
[tex]A + B =\left[\begin{array}{ccc}1&2\\3&4\\\end{array}\right] + \left[\begin{array}{ccc}5&6\\7&8\\\end{array}\right][/tex]
[tex]A + B = \left[\begin{array}{ccc}1+5 &2+6 &\\3 + 7&4 + 8\\\end{array}\right][/tex]
[tex]A + B =\left[\begin{array}{ccc}6&8\\10&12\\\end{array}\right][/tex]
[tex](A + B)^{T} = \left[\begin{array}{ccc}6&10\\8&12\\\end{array}\right][/tex] ------(1)
2. Solving for [tex]A^{T} + B^{T}[/tex]
[tex]A^{T} + B^{T} = \left[\begin{array}{ccc}1&3\\2&4\\\end{array}\right] + \left[\begin{array}{ccc}5&7\\6&8\\\end{array}\right][/tex]
[tex]A^{T} + B^{T} = \left[\begin{array}{ccc}1 + 5&3 +7\\2 + 6&4+8\\\end{array}\right][/tex]
[tex]A^{T} + B^{T} = \left[\begin{array}{ccc}6&10\\8&12\\\end{array}\right][/tex] -----(2)
Comparing (1) and (2)
[tex](A + B)^{T} = A^{T} + B^{T}[/tex]
