Given, A + B = 2B^T .......(1)

$$ \Rightarrow $$ (A + B)^T = (2B^T)^T

$$ \Rightarrow $$ A^T + B^T = 2B

$$ \Rightarrow $$ B = $${{{A^T} + {B^T}} \over 2}$$

Now put this in equation (1)

So, A + $${{{A^T} + {B^T}} \over 2}$$ = 2B^T

$$ \Rightarrow $$2A + A^T = 3B^T

$$ \Rightarrow $$ 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

$$ \Rightarrow $$ 11B^T - A^T = 2I₃

$$ \Rightarrow $$ (11B^T - A^T)^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