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 (3 p - 3 s/5 q - 4 r) is equal to

Question

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 (3 p - 3 s/5 q - 4 r) is equal to (A) (3/2) (B) 4 (C) (9/2) (D) 5

Tardigrade Admin · Accepted Answer

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 (3 (p - s)/5 q - 4 r)=(3 (- r)/5 ((2 r)3) - 4 r)=(- 3/(10)3 - 4)=(9/2)=4.5

Q. Let $A = [1324]$ and $B = [p r q s]$ 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

$\frac{3}{2}$

$4$

$\frac{9}{2}$

$5$

Q. Let A=[13​24​] and B=[pr​qs​] are two matrices such that AB=BA and r=0 , then the value of 5q−4r3p−3s​ is equal to

23​

4

29​

5

AB=[13​24​][pr​qs​]=[p+2r3p+4r​q+2s3q+4s​] BA=[pr​qs​][13​24​]=[p+3qr+3s​2p+4q2r+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 5q−4r3(p−s)​=5(32r​)−4r3(−r)​=310​−4−3​=29​=4.5

Q. Let $A = [1324]$ and $B = [p r q s]$ 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

$\frac{3}{2}$

$4$

$\frac{9}{2}$

$5$