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3 and 4 .Determinants and Matrices
hard
અહી $A=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha>0$, આપેલ છે કે જેથી $\operatorname{det}(A)=0$ અને $\alpha+\beta=1$ છે. જો $I$ એ $2 \times 2$ એકમ શ્રેણિક હોય તો શ્રેણિક $(I+ A )^8 = $
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$\left[\begin{array}{cc}1025 & -511 \\ 2024 & -1024\end{array}\right]$
D$\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$
(JEE MAIN-2025)
Solution
$|A|=0$
$\alpha \beta+6=0$
$\alpha \beta=-6$
$\alpha+\beta=1$
$\Rightarrow \alpha=3, \beta=-2$
$A=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]$
$A^2=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]$
$\therefore A^2=A$
$A= A ^2= A ^3= A ^4= A ^5$
$( I + A )^8$
$= I +{ }^8 C _1 A^7+{ }^8 C _2 A^6+\ldots .+{ }^8 C _8 A^8$
$= I + A \left({ }^8 C _1+{ }^8 C _2+\ldots . .+{ }^8 C _8\right)$
$= I + A \left(2^8-1\right)$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]+\left[\begin{array}{cc}765 & -255 \\ 1530 & -510\end{array}\right]$
$=\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$
$\alpha \beta+6=0$
$\alpha \beta=-6$
$\alpha+\beta=1$
$\Rightarrow \alpha=3, \beta=-2$
$A=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]$
$A^2=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]=\left[\begin{array}{ll}3 & -1 \\ 6 & -2\end{array}\right]$
$\therefore A^2=A$
$A= A ^2= A ^3= A ^4= A ^5$
$( I + A )^8$
$= I +{ }^8 C _1 A^7+{ }^8 C _2 A^6+\ldots .+{ }^8 C _8 A^8$
$= I + A \left({ }^8 C _1+{ }^8 C _2+\ldots . .+{ }^8 C _8\right)$
$= I + A \left(2^8-1\right)$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]+\left[\begin{array}{cc}765 & -255 \\ 1530 & -510\end{array}\right]$
$=\left[\begin{array}{cc}766 & -255 \\ 1530 & -509\end{array}\right]$
Standard 12
Mathematics
