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 $\left[\begin{array}{cc}1& 2500\\ 0& 1\end{array}\right]$

$\prod ^{50}\left[\begin{array}{cc}1& 2r-1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]$.....

$=\left[\begin{array}{cc}1& 1+3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 5\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 7\\ 0& 1\end{array}\right]$.......

$=\left[\begin{array}{cc}1& 1+3+5\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 7\\ 0& 1\end{array}\right]$........

$=\left[\begin{array}{cc}1& 1+3+\mathrm{5....}+T_{50}\\ 0& 1\end{array}\right]$

$1+3+5+......+T_{50}=\frac{50}{2}\left(2\right(1)+49\times 2)=2500$

$\therefore$ Option $D$ is correct