Let I= ( array ll 1 & 0 \\ 0 & 1 array ) and P= ( array ll 2 & 0 \\ 0 & 3 array ). Let Q= ( array ll x & y \\ z & 4 array ) for some non-zero real numbers x, y, and z, for which there is 2 2 matrix R with all entries being non-zero real numbers, such that Q R=R P. Then which of the following statements is (are) TRUE?
B. The determinant of Q -6 I is 12
aligned & QR = RP \\ & P= ( array ll 2 & 0 \\ 0 & 3 array ) Q= ( array ll x & y \\ z & 4 array ) \\ & ( array ll x & y \\ z & 4 array ) ( array ll r_1 & r_2 \\ r_3 & r_4 array )= ( array ll r_1 & r_2 \\ r_3 & r_4 array ) ( array ll 2 & 0 \\ 0 & 3 array ) \\ & . array l xr _1+ yr _3=2 r _1 \\ zr _1+4 r _3=2 r _3 array \ x -2 - y = r _3 r _1 = z -2 aligned \(2 x-4=y z\) . array l xr _2+ yr _4=3 r _2 \\ zr _2+4 r _4=3 r _4 array \ x -3 - y = r _4 r _2 =- z \( x -3= yz \) 2 x-4=x-3 \( x=1 ~\&~ y z=-2\) aligned & Q - I = ( array cc x - & y \\ z & 4- array ) \\ & | Q - I |=( - x )( -4)- yz \\ & = ^2-( x +4) +4 x - yz \\ & | Q - I |= ^2-5 +6 aligned Now verify the option
Related: Mathematics — Matrices · All PYQ Banks