Mathematics - Matrices Question with Solution | TestHub

Given$A+B=2B^T$

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

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

$=A+\left(\frac{B^T+A^T}{2}\right)=2B^T$

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

$\Rightarrow A=\frac{3B^T-A^T}{2}$

$3A+2B=I_3$......$\left(i\right)$

$\Rightarrow 3\left(\frac{3B^T-A^T}{2}\right)+2\left(\frac{A^T+B^T}{2}\right)=I_3$

$\Rightarrow \left(\frac{3B^T+2B^T}{2}\right)+2\left(\frac{A^T+B^T}{2}\right)=I_3$

$\Rightarrow 11B^T-A^T=2I_3$.......$\left(ii\right)$

Equation$\left(i\right)+\left(ii\right)$

$35B=7I_3$

$\Rightarrow B=\frac{I_3}{5}$

$11 \frac{I_3}{5}-A=2I_3$

$\Rightarrow 11\frac{I_3}{5}-2I_3=A$

$\Rightarrow A=\frac{I_3}{5}$

$\because 5A=5B=I_3$

$\Rightarrow 10A+5B=3I_3$