Question

Let $A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B=\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ are two matrices such that $AB=BA$ and $r\neq 0$ , then the value of $\frac{3 p - 3 s}{5 q - 4 r}$ is equal to

• A. $\frac{3}{2}$
• B. $4$
• C. $\frac{9}{2}$
• D. $5$

Answer 1

Answer: (C)

$AB=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} p & q \\ r & s \end{bmatrix}=\begin{bmatrix} p+2r & q+2s \\ 3p+4r & 3q+4s \end{bmatrix}$
$BA=\begin{bmatrix} p & q \\ r & s \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} p+3q & 2p+4q \\ r+3s & 2r+4s \end{bmatrix}$
$\because AB=BA\Rightarrow p+2r=p+3q\Rightarrow 2r=3q$
$\Rightarrow q+2s=2p+4q\Rightarrow 2s=2p+3q$
$\Rightarrow 3p+4r=r+3s\Rightarrow p+r=s\Rightarrow p-s=-r$
$\Rightarrow 3q+4s=2r+4s\Rightarrow 2r=3q$
$\frac{3 \left(p - s\right)}{5 q - 4 r}=\frac{3 \left(- r\right)}{5 \left(\frac{2 r}{3}\right) - 4 r}=\frac{- 3}{\frac{10}{3} - 4}=\frac{9}{2}=4.5$