Question

Consider $A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}$ such that $AB=BA$ , then the value of $\frac{a_{12}}{a_{21}}+\frac{a_{11}}{a_{22}}$ is

• A. $2$
• B. $4$
• C. $\frac{3}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (C)

Given:
$AB=BA$
$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$
$\begin{bmatrix} a_{11}+2a_{12} & a_{11}+a_{12} \\ a_{21}+2a_{22} & a_{21}+a_{22} \end{bmatrix}=\begin{bmatrix} a_{11}+a_{21} & a_{12}+a_{22} \\ 2a_{11}+a_{21} & 2a_{12}+a_{22} \end{bmatrix}$
$a_{11}+2a_{12}=a_{11}+a_{21}$ $\Rightarrow 2a_{12}=a_{21}$
$a_{11}+a_{12}=a_{12}+a_{22}\Rightarrow a_{11}=a_{22}$
$a_{21}+2a_{22}=2a_{11}+a_{21}\Rightarrow a_{11}=a_{22}$
$a_{21}+a_{22}=2a_{12}+a_{22}\Rightarrow a_{21}=2a_{12}$
$\frac{a_{12}}{a_{21}}=\frac{1}{2},\frac{a_{11}}{a_{22}}=1$