Let A= [ cc 1 & 1 51 0 & 1 ]. If B= [ cc 1 & 2 -1 & -1 ] A [ cc -… - Vidyadip

MCQ

Let $A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$. If $B=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A \left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$ then the sum of all the elements of the matrix $\sum \limits_{n=1}^{50} B^n$ is equal to

✓
$100$
B
$50$
C
$75$
D
$125$

✓

Answer

Correct option: A.

$100$

a
Let $C=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right], D =\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$

$DC =\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right]\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]= I$

$\begin{aligned} & B = CAD \\ & B ^{ n }=\underbrace{( CAD )( CAD )( CAD ) \ldots( CAD )}_{ n -\text { times }}\end{aligned}$

$\Rightarrow B ^{ n }= CA ^{ n } D$

$A ^2=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1 & \frac{2}{51} \\ 0 & 1\end{array}\right]$

$A^3=\left[\begin{array}{cc}1 & \frac{3}{51} \\ 0 & 1\end{array}\right]$

similarly $A^{ n }=\left[\begin{array}{cc}1 & \frac{ n }{51} \\ 0 & 1\end{array}\right]$

$B ^{ n }=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right]\left[\begin{array}{cc}1 & \frac{ n }{51} \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$

$=\left[\begin{array}{cc}1 & \frac{ n }{51}+2 \\ -1 & -\frac{ n }{51}-1\end{array}\right]\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$

$=\left[\begin{array}{cc}\frac{ n }{51}+1 & \frac{ n }{51} \\ -\frac{ n }{51} & 1-\frac{ n }{51}\end{array}\right]$

$\sum \limits_{ n =1}^{50} B ^{ n }=\left[\begin{array}{cc}25+50 & 25 \\ -25 & -25+50\end{array}\right]=\left[\begin{array}{cc}75 & 25 \\ -25 & 25\end{array}\right]$

Sum of the elements $=100$

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