Question

For a matrix $A=\left[\begin{array}{cc}1& 2r-1\\ 0& 1\end{array}\right]$, the value of $\prod _{r=1}^{50}\left[\begin{array}{cc}1& 2r-1\\ 0& 1\end{array}\right]$ is equal to-

• A. $\left[\begin{array}{cc}1& 100\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 4950\\ 0& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 5050\\ 0& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& 2500\\ 0& 1\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{cc}1& 2500\\ 0& 1\end{array}\right]$

$\begin{array}{c}A=\left[\begin{array}{cc}1& 2r-1\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]..............\left[\begin{array}{cc}1& 99\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}1& 1\times 3+1\times 1\\ 0& 0\end{array}\right]\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}1& 1\times 5+1\times 3+1\times 1\\ 0& 0\end{array}\right]\left[\begin{array}{cc}1& 7\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 9\\ 0& 1\end{array}\right]..........\left[\begin{array}{cc}1& 99\\ 0& 1\end{array}\right]\end{array}$

Similarly,

$\begin{array}{c}=\left[\begin{array}{cc}1& 1\times 1+1\times 3+1\times 5+............+1\times 99\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}1& {\sum }_{r=1}^{50}\left(2r-1\right)\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 2500\\ 0& 1\end{array}\right]\\ ={\sum }_{r=1}^{50}\left(2r-1\right)=\frac{2\times 50\times \left(51\right)}{2}-50\\ =2500\end{array}$

Hence,

Option $D$ is correct answer.