Let B= [ ll 1 & 3 1 & 5 ] and A be a 2 2 matrix such that AB^ -1 … - Vidyadip

MCQ

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $\mathrm{AB}^{-1}=\mathrm{A}^{-1}$. If $\mathrm{BCB}^{-1}=\mathrm{A}$ and $\mathrm{C}^4+\alpha \mathrm{C}^2+\beta \mathrm{I}=\mathrm{O}$, then $2 \beta-\alpha$ is equal to :

A
$16$
B
$2$
C
$8$
✓
$10$

✓

Answer

Correct option: D.

$10$

d
$ \mathrm{BCB}^{-1}=\mathrm{A} $

$ \Rightarrow\left(\mathrm{BCB}^{-1}\right)\left(\mathrm{BCB}^{-1}\right)=\mathrm{A} \cdot \mathrm{A} $

$ \Rightarrow \mathrm{BCI} \mathrm{CB^{-1 } = \mathrm { A } ^ { 2 }} $

$ \Rightarrow \mathrm{BC}^2 \mathrm{~B}^{-1}=\mathrm{A}^2 $

$ \Rightarrow \mathrm{B}^{-1}\left(\mathrm{BC}^2 \mathrm{~B}^{-1}\right) \mathrm{B}=\mathrm{B}^{-1}(\mathrm{~A} \cdot \mathrm{A}) \mathrm{B}$

From equation $(1)$

$ \mathrm{C}^2=\mathrm{A}^{-1} \cdot \mathrm{A} \cdot \mathrm{B} $

$ \mathrm{C}^2=\mathrm{B} $

$ \text { Also } \mathrm{AB}^{-1}=\mathrm{A}^{-1} $

$ \Rightarrow \mathrm{AB}^{-1} \cdot \mathrm{A}=\mathrm{A}^{-1} \mathrm{~A}=\mathrm{I} $

$ \Rightarrow \mathrm{A}^{-1}\left(A \mathrm{~B}^{-1} \mathrm{~A}\right)=\mathrm{A}^{-1} \mathrm{I} $

$ \mathrm{B}^{-1} \mathrm{~A}=\mathrm{A}^{-1}$

Now characteristics equation of $\mathrm{C}^2$ is

$ \left|C_2-\lambda I\right|=0 $

$ |B-\lambda I|=0$

$\Rightarrow\left|\begin{array}{cc}1-\lambda & 3 \\ 1 & 5-\lambda\end{array}\right|=0$

$ \Rightarrow(1-\lambda)(5-1)-3=0 \Rightarrow\left(\lambda^2-6 \lambda+5\right)-3=0 $

$ \Rightarrow \lambda^2-6 \lambda+2=0 $

$ \Rightarrow \beta^2-6 B+2 I=0 $

$ \Rightarrow C^4-6 C^2+2 I=0 $

$ \alpha=-6 $

$ \beta=2 $

$ \therefore 2 \beta-\alpha=4+6=10$

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