Let I=\left(\begin{array}{cc}1 & 0 \ 0 & 1)\end{array} and P=\left(\begin{array}{cc}2 & 0 \ 0 & 3)\end{array}. Let Q=\left(\begin{array}{cc}x & y \ z & 4)\end{array} for some non-zero real numbers x, y, and z, for which there is a 2 \times 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? [JEE Adv 2025 P2]

The determinant of Q-2 I is zero

Matrices and Determinants

Section: Mathematics

Question : 2 of 56

Marks: +1, -0

Let I=(

1	0
0	1

) and P=(

2	0
0	3

). Let Q=(

x	y
z	4

)
for some non-zero real numbers x,y, and z, for which there is a 2×2 matrix R with all entries being non-zero real numbers, such that QR=RP.
Then which of the following statements is (are) TRUE?

[JEE Adv 2025 P2]

The determinant of Q−2I is zero
The determinant of Q−6I is 12
The determinant of Q−3I is 15
yz=2

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