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माना $A =\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$ तथा $B =\left[\begin{array}{ll}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \beta \in R$ हैं। माना $( A + B )^2= A ^2+\left[\begin{array}{ll}2 & 2 \\ 2 & 2\end{array}\right]$ को संतुष्ट करने वाला $\alpha$ का मान $\alpha_1$ है तथा $( A + B )^2= B ^2$ को संतुष्ट करने वाला $\alpha$ का मान $\alpha_2$ हैं। तब $\left|\alpha_1-\alpha_2\right|$ बराबर है $...........$ I
$2$
$22$
$3$
$8$
Solution
$A+B=\left[\begin{array}{cc}\beta+1 & 0 \\ 3 & \alpha\end{array}\right]$
$(A+B)^{2}=\left[\begin{array}{cc}\beta+1 & 0 \\ 3 & \alpha\end{array}\right]\left[\begin{array}{cc}\beta+1 & 0 \\ 3 & \alpha\end{array}\right]$
$=\left[\begin{array}{cc}(\beta+1)^{2} & 0 \\ 3(\beta+1)+3 \alpha & \alpha^{2}\end{array}\right]$
$A^{2}=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$
$=\left[\begin{array}{cc}-1 & -1-\alpha \\ 2+2 \alpha & \alpha^{2}-2\end{array}\right]$
$\therefore\left[\begin{array}{cc}1 & -\alpha+1 \\ 2 \alpha+4 & \alpha^{2}\end{array}\right]=\left[\begin{array}{cc}(\beta+1)^{2} & 0 \\ 3(\alpha+\beta+1) & \alpha^{2}\end{array}\right]$
$\alpha=1]=\alpha_{1}$
$B ^{2}=\left[\begin{array}{ll}\beta & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}\beta & 1 \\ 1 & 0\end{array}\right]$
$=\left[\begin{array}{cc}\beta^{2}+1 & \beta \\ \beta & 1\end{array}\right]=\left[\begin{array}{cc}(\beta+1)^{2} & 0 \\ 3(\beta+1)+3 \alpha & \alpha^{2}\end{array}\right]$
$\therefore \beta=0, \alpha=-1=\alpha_{2}$
$\left|\alpha_{1}-\alpha_{2}\right|=|1-(-1)|=2$
