Let $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ and $P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$. Let $Q=\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ \mathrm{z} & 4\end{array}\right)$ 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?

(A) The determinant of $Q-2 I$ is zero; (B) The determinant of $\mathrm{Q}-6 I$ is 12

(C) The determinant of $Q-3 I$ is 15

JEE Advanced · 2025 · 18 May 2025 · Paper 2 · Matrices

Let

I = (1001)

and

P = (2003)

. Let

Q = (x z y 4)

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

QR = R P

.
Then which of the following statements is (are) TRUE?

Answer: (A), (B)

From

QR = R P

, comparing entries gives two conditions:

From these:

2 x - 4 = x - 3

, giving

x = 1

and

y z = - 2

.

So

Q = (1 z y 4)

with

y z = - 2

.

The characteristic polynomial is:

∣ Q - λ I ∣ = λ^{2} - (x + 4) λ + (4 x - y z) = λ^{2} - 5 λ + 6 = (λ - 2) (λ - 3)

Checking options:

JEE Advanced · 2025 · 18 May 2025 · Paper 2 · One or more correct options