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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?

Options

  1. AThe determinant of Q-2 I is zero
  2. BThe determinant of Q -6 I is 12
  3. CThe determinant of Q-3 I is 15
  4. Dy z=2

Correct answer

A. The determinant of Q-2 I is zero

Step-by-step solution

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 r₁ & r₂ r₃ & r₄ array )= ( array ll r₁ & r₂ r₃ & r₄ 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

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