Let A= [ cc & -1 6 & ], >0, such that det(A)=0 and + =1. If I den… - Vidyadip

MCQ

Let $A=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha>0$, such that $\operatorname{det}(A)=0$ and $\alpha+\beta=1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(1+ A )^8$ is:

A
$\left[\begin{array}{ll}4 & -1 \\ 6 & -1\end{array}\right]$
B
$\left[\begin{array}{cc}257 & -64 \\ 514 & -127\end{array}\right]$
C
$\left[\begin{array}{cc}1025 & -511 \\ 2024 & -1024\end{array}\right]$
✓
$\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$

✓

Answer

Correct option: D.

$\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$

(D) $\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$
Explanation:
$\begin{array}{l}|A|=0 \\
\alpha \beta+6=0 \\
\alpha \beta=-6 \\
\alpha+\beta=1\end{array}$
$\Rightarrow \alpha=3, \beta=-2$
$A=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]$
$A^2=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]$
$\begin{array}{l}\therefore A ^2= A
\\ A = A ^2= A ^3= A ^4= A ^5\end{array}$
$( I + A )^8$
$= I +{ }^8 C _1 A^7+{ }^8 C _2 A^6+\ldots . .+{ }^8 C _8 A^8$
$= I + A \left({ }^8 C _1+{ }^8 C _2+\ldots . .+{ }^8 C _8\right)$
$= I + A \left(2^8-1\right)$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]+\left[\begin{array}{cc}765 & -255 \\ 1530 & -510\end{array}\right]$
$=\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$

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