Let the matrix A= [ array lll 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 array ] satisfy A^n=A^ n-2 +A^2-I for n ≥ 3 . Then the sum of all the elements of A^ 50 is :-

Correct answer: A. 53 aligned & A^ 50 =A^ 48 +A^2-I & =A^ 46 +2(~A^2-I) & =A^ 44 +3(~A^2-I) & =A^2+24(~A^2-I) & =25 ~A^2-24 I aligned aligned & =25[ array lll 1 & 0 & 0 1 & 1 & 0 1 & 0 & 1 array ]-24[ array lll 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 array ] & =[ array ccc 1 & 0 & 0 25 & 1 & 0 25 & 0 & 1 array ] aligned Sum =53 option (1)

JEE Main 20254 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 4 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2025 · 4 April · Evening Shift · Mathematics · Matrices

Let the matrix A= [ array lll 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 array ]...

Let the matrix

A=

\left[

\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}

\right]

satisfy

A^n=A^{n-2}+A^2-I

for

\mathrm{n} \geq 3

. Then the sum of all the elements of

\mathrm{A}^{50} is :-

Options

A $53$
B $52$
C $39$
D $44$

Correct answer

A. $53$

Step-by-step solution

\begin{aligned} & \mathrm{A}^{50}=\mathrm{A}^{48}+\mathrm{A}^2-\mathrm{I} \\ & =\mathrm{A}^{46}+2\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =\mathrm{A}^{44}+3\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =\mathrm{A}^2+24\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =25 \mathrm{~A}^2-24 \mathrm{I}\end{aligned}

\begin{aligned}<br/>& =25\left[\begin{array}{lll}<br/>1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1<br/>\end{array}\right]-24\left[\begin{array}{lll}<br/>1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br/>\end{array}\right] \\ & =\left[\begin{array}{ccc}<br/>1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1<br/>\end{array}\right]<br/>\end{aligned}

Sum

=53

option (1)

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