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 Q R = R P . Then which of the following

Question

Let $I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ and $P = (\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}) .$ Let $Q = (\begin{matrix} x & y \\ z & 4 \end{matrix})$ 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 statement(s) is (are) TRUE [2025]

Then which of the following statement(s) is (are) TRUE? [2025]

Smart Achievers · Accepted Answer

(1, 2)

$Given: Q R = R P \Rightarrow Q R - 2 R = R P - 2 R$

$\Rightarrow$ $| Q - 2 I |$ $| R |$ $=$ $| R |$ $| \begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix} |$ $=$ $| R |$ $\cdot 0$

$\Rightarrow$ $| Q - 2 I |$ $= 0$ $\dots (i)$

Also, $Q R - 6 R = R P - 6 R$

$\Rightarrow$ $| Q - 6 I |$ $| R |$ $=$ $| R |$ $| \begin{matrix} - 4 & 0 \\ 0 & - 3 \end{matrix} |$ $=$ $| R |$ $\cdot 12$

So, $| Q - 6 I |$ $= 12$ $\dots (i i)$

Now, $Q R - 3 R = R P - 3 R$

$\Rightarrow$ $| Q - 3 I |$ $| R |$ $=$ $| R |$ $| \begin{matrix} - 1 & 0 \\ 0 & 0 \end{matrix} |$ $=$ $| R |$ $\cdot 0$

So, $| Q - 3 I |$ $= 0$ $\dots (i i i)$

Now, from (i), $| Q - 2 I |$ $= 0$

$\Rightarrow$ $| \begin{matrix} x - 2 & y \\ z & 2 \end{matrix} |$ $= 0$ $\Rightarrow 2 x - 4 = y z$ $\dots (i v)$

From (iii), $| Q - 3 I |$ $= 0$

$\Rightarrow$ $| \begin{matrix} x - 3 & y \\ z & 1 \end{matrix} |$ $= 0$ $\Rightarrow x - 3 = y z$ $\dots (v)$

From (iv) and (v), $2 x - 4 = x - 3$

$\Rightarrow x = 1,$

so, $y z = - 2$

Solution

1 The determinant of $Q - 2 I$ is zero

2 The determinant of $Q - 6 I$ is 12

3 The determinant of $Q - 3 I$ is 15

4 $y z = 2$

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Solution

Q. Let I=(1001) and P=(2003). Let Q=(xyz4) 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 statement(s) is (are) TRUE [2025]

1 The determinant of Q−2I is zero

2 The determinant of Q−6I is 12

3 The determinant of Q−3I is 15

4 yz=2

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1 The determinant of $Q - 2 I$ is zero

2 The determinant of $Q - 6 I$ is 12

3 The determinant of $Q - 3 I$ is 15

4 $y z = 2$