Let A = [ cases 1 & 0 & 0 1 & 1 & 0 1 & 1 & 1 cases ] and B = A ^ 20 . Then the sum of the elements of the first column of B is

Correct answer: D. 231 Given, A = [ cases 1 & 0 & 0 1 & 1 & 0 1 & 1 & 1 cases ] ⇒ A ^ 2 = [ cases 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 cases ]; A ^ 3 = [ cases 1 & 0 & 0 3 & 1 & 0 6 & 3 & 1 cases ] A ^ 4 = [ cases 1 & 0 & 0 4 & 1 & 0 10 & 4 & 1 cases ]; , ; A ^ 20 = [ cases 1 & 0 & 0 20 & 1 & 0 210 & 20 & 1 cases ](∵ a _ 31 = 20thtermin1 + 3 + 6 + 10 + . . . .) Hence, sum of the elements of first column = 231 .

JEE Main 201816 AprilOnline

Exam: JEE Main
Year: 2018
Date: 16 April
Mode: Online
Subject: Mathematics
Chapter: Matrices

JEE Main 2018 · 16 April · Online · Mathematics · Matrices

Let A = [ cases 1 & 0 & 0 1 & 1 & 0 1 & 1 & 1 cases ] and B = A ^...

Let

A = \left[

\begin{cases}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{cases}

\right] and

B = {A}^{20}

. Then the sum of the elements of the first column of

B$ is

Options

A $210$
B $211$
C $251$
D $231$

Correct answer

D. $231$

Step-by-step solution

Given, $A = \left[$ \begin{cases}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{cases} $\right]$

$⇒$ ${A}^{2} = \left[$ \begin{cases}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{cases} $\right]; {A}^{3} = \left[$ \begin{cases}1 & 0 & 0 \\ 3 & 1 & 0 \\ 6 & 3 & 1\end{cases} $\right]$

${A}^{4} = \left[$ \begin{cases}1 & 0 & 0 \\ 4 & 1 & 0 \\ 10 & 4 & 1\end{cases} $\right]; , \ldots ; {A}^{20} = \left[$ \begin{cases}1 & 0 & 0 \\ 20 & 1 & 0 \\ 210 & 20 & 1\end{cases} $\right]\left(∵ {a}_{31} = 20thtermin1 + 3 + 6 + 10 + . . . .\right)$

Hence, sum of the elements of first column $= 231$ .

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