Question

For two $3\times 3$ matrices $A$ and $B$, let $A+B=2B^{\prime }$ and $3A+2B=I_3,$ where $B^{\prime }$ is the transpose of $B$ and $I_3$ is $3\times 3$ identity matrix. Then :

• A. $5A+10B=2I_3$
• B. $10A+5B=3I_3$
• C. $B+2A=I_3$
• D. $3A+6B=2I_3$

Answer 1

Answer: (B)

$A^T+B^T=2B$
$\Rightarrow A+\left(\frac{B^T+A^T}{2}\right)=2B^T$
$2A+A^T=2B^T$
$\Rightarrow A=\frac{3B^T-A^T}{2}$
$3A+2B=I_3\dots (i)$
$\Rightarrow 3\left(\frac{3B^T-A^T}{2}\right)+2\left(\frac{A^T+B^T}{2}\right)=I_3$
$\Rightarrow \left(\frac{3B^T+2B^T}{2}\right)+\left(\frac{2A^T-3A^T}{2}\right)=I_3$
$\Rightarrow 11B^T-A^T=2I_3\dots (ii)$
Equation $(i)+(ii)$
$35B=7I_3$
$\Rightarrow B=\frac{I_3}{5}$
$11\frac{I_3}{5}A=2I_3$
$\Rightarrow 11\frac{I_3}{5}-2I_3=A$
$\Rightarrow A=\frac{I_3}{4}$
$\because 5A=5B=I_3$
$\Rightarrow 10A+5B=3I_3$