Official solution
Given, A + B = 2BT .......(1)
\[ \Rightarrow \] (A + B)T = (2BT)T
\[ \Rightarrow \] AT + BT = 2B
\[ \Rightarrow \] B = \[{{{A^T} + {B^T}} \over 2}\]
Now put this in equation (1)
So, A + \[{{{A^T} + {B^T}} \over 2}\] = 2BT
\[ \Rightarrow \]2A + AT = 3BT
\[ \Rightarrow \] A = \[{{3{B^T} - {A^T}} \over 2}\]
Also, 3A + 2B = I3 .......(2)
\[ \Rightarrow \] \[3\left( {{{3{B^T} - {A^T}} \over 2}} \right) + 2\left( {{{{A^T} + {B^T}} \over 2}} \right)\] = I3
\[ \Rightarrow \] 11BT - AT = 2I3
\[ \Rightarrow \] (11BT - AT)T = (2I3)T
\[ \Rightarrow \] 11B - A = 2I3 ........(3)
Multiply (3) by 3 and then adding (2) and (3) we get,
35B = 7I3
\[ \Rightarrow \] B = \[{{{I_3}} \over 5}\]
From (3), 11\[{{{I_3}} \over 5}\] - A = 2I3
\[ \Rightarrow \] A = \[{{{I_3}} \over 5}\]
\[ \therefore \] 5A = 5B = I3
\[ \Rightarrow \] 10A + 5B = 3I3