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  • Let <img alt="\mathrm{S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmat

Let \mathrm{S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}} and let \mathrm{T_{n}=\left\{A \in S: A^{n(n+1)}=I\right\}}. Then the number of elements in \mathrm{\cap_{n=1}^{100} T_{n}} is

Option: 1

100


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{ S=\left[\begin{array}{cc} -1 & a \\ 0 & b \end{array}\right] ; \quad a, b \in\{1,2,3, \ldots 100\} }

\mathrm{ T_{n}=\left\{A \in s: A^{n(n+1)}=I\right\} }

\mathrm{ A^{2}=\left[\begin{array}{cc} -1 & a \\ 0 & b \end{array}\right]\left[\begin{array}{cc} -1 & a \\ 0 & b \end{array}\right]=\left[\begin{array}{cc} 1 & -a+a b \\ 0 & b^{2} \end{array}\right] }

b should be 1

\mathrm{ A^{2}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] }  hence a can take any value 

\mathrm{\therefore } Total number of elements = 100

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HARSH KANKARIA

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