Let and let , and . Then, which of the following options is/are…

MATHEMATICS · Single correct · CBSE · Mathematics · Matrices · Matrices

Problem

Let

x \in \mathbb{R}

and let

P = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{bmatrix}

Q = \begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{bmatrix}

and

R = PQP^{-1}

. Then, which of the following options is/are correct? (A)

\det R = \det \begin{bmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 5 \end{bmatrix} + 8

\forall x \in \mathbb{R}

. (B) For

x=1

\exists

a unit vector

\hat{\alpha} + \hat{\beta} + x\hat{k}

for which

R \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

. (C) For

x=0

, if

R \begin{bmatrix} 1 \\ a \\ b \end{bmatrix} = 6 \begin{bmatrix} 1 \\ a \\ b \end{bmatrix}

, then

a+b=5

. (D) There exists a real number

x

such that

PQ=QP