Let A= 1 2 3 4 and B= p q r s are two matrices such that AB=BA and r≠0, then the value of (3p-3s)/(5q-4r) is equal to

Correct answer: (C) 9/2. AB= 1 2 3 4 p q r s = p+2r q+2s 3p+4r 3q+4s BA= p q r s 1 2 3 4 = p+3q 2p+4q r+3s 2r+4s ∵AB=BA⇒p+2r=p+3q⇒2r=3q ⇒q+2s=2p+4q⇒2s=2p+3q ⇒3p+4r=r+3s⇒p+r=s⇒p-s=-r ⇒3q+4s=2r+4s⇒2r=3q (3p-s)/(5q-4r)=(3-r)/52r3-4r=(-3)/103-4=9/2=4.5

Q6 / 75 · NTA Abhyas (JEE Main) 2020 · Mathematics · Matrices

Let $A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ and $B = [\begin{matrix} p & q \\ r & s \end{matrix}]$ are two matrices such that $A B = B A$ 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
$4$
Correct answer
C
$\frac{9}{2}$
D
$5$

Solution

A B = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} p & q \\ r & s \end{matrix}] = [\begin{matrix} p + 2 r & q + 2 s \\ 3 p + 4 r & 3 q + 4 s \end{matrix}]

B A = [\begin{matrix} p & q \\ r & s \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] = [\begin{matrix} p + 3 q & 2 p + 4 q \\ r + 3 s & 2 r + 4 s \end{matrix}]

∵ A B = B A \Rightarrow p + 2 r = p + 3 q \Rightarrow 2 r = 3 q

\Rightarrow q + 2 s = 2 p + 4 q \Rightarrow 2 s = 2 p + 3 q

\Rightarrow 3 p + 4 r = r + 3 s \Rightarrow p + r = s \Rightarrow p - s = - r

\Rightarrow 3 q + 4 s = 2 r + 4 s \Rightarrow 2 r = 3 q

\frac{3 (p - s)}{5 q - 4 r} = \frac{3 (- r)}{5 (\frac{2 r}{3}) - 4 r} = \frac{- 3}{\frac{10}{3} - 4} = \frac{9}{2} = 4.5

Hence, the correct answer is option: 'C'.

Let A=1234 and B=pqrs are two matrices such that AB=BA and r≠0, then the value of 3p-3s5q-4r is equal to

Solution

More Matrices Questions

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