Let A = [ 1 1 51 0 1 ] . If B = [ 1 2 - 1 - 1 ] A [ - 1 - 2 1 1 ] , then the sum of all the elements of the matrix n = 1 50 B n is equal to [2023]

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Solution

Q.
Let $A = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] .$ If $B = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] A [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}],$ then the sum of all the elements of the matrix $\sum_{n = 1}^{50} B^{n}$ is equal to [2023]

1 50

2 100

3 75

4 125

Ans.
(2)

$A = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}]; B = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] A [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

Let $P = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}], Q = [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$B = P A Q$

$B^{2} = (P A Q) (P A Q) = P A Q P A Q = P A^{2} Q$

As, $Q P = [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$A^{2} = [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & \frac{2}{51} \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$A^{3} = [\begin{matrix} 1 & \frac{2}{51} \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & \frac{1}{51} \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & \frac{3}{51} \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

Similarly, $A^{n} = [\begin{matrix} 1 & \frac{n}{51} \\ 0 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$B^{n} = P A^{n} Q = [\begin{matrix} 1 & 2 \\ - 1 & - 1 \end{matrix}] [\begin{matrix} 1 & \frac{n}{51} \\ 0 & 1 \end{matrix}] [\begin{matrix} - 1 & - 2 \\ 1 & 1 \end{matrix}] \begin{matrix} \end{matrix}$

$B^{n} = [\begin{matrix} 1 + \frac{n}{51} & \frac{n}{51} \\ \frac{- n}{51} & 1 - \frac{n}{51} \end{matrix}] \begin{matrix} \end{matrix}$

$\sum_{n = 1}^{50} B^{n} = [\begin{matrix} 50 + 25 & 25 \\ - 25 & 50 - 25 \end{matrix}] = [\begin{matrix} 75 & 25 \\ - 25 & 25 \end{matrix}] \begin{matrix} \end{matrix}$

Sum of the elements = 100

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