QJEEM17Ma18e8: JEE Main 2017 Mathematics - Class 12 JEEM17Ma18e8 Solution: For two 3 × 3 matrices A and B, let A + B = 2B T and 3A + 2B =…

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