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माना $A =\left[\begin{array}{ll} x & 1 \\ 1 & 0\end{array}\right], x \in R$ तथा $A ^{4}=\left[ a _{ ij }\right]$ है। यदि $a _{11}$ $=109$ है, तो $a _{22}$ बराबर है |
$10$
$-8$
$-10$
$8$
Solution
$A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$
$A^{2}=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]$
$A^{4}=\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]$
$=\left[\begin{array}{ll}\left(x^{2}+1\right)^{2}+x^{2} & x\left(x^{2}+1\right)+x \\ x\left(x^{2}+1\right)+x & x^{2}+1\end{array}\right]$
$a_{11}=\left(x^{2}+1\right)^{2}+x^{2}=109$
$\Rightarrow x =\pm 3$
$a_{22}=x^{2}+1=10$
