Let the matrix A= [begin array lll 1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0end array ] satisfy A n =A n-2 +A 2 -l for n geq 3 . Then the sum of all the elements of A 50 is:

Mathematics · Matrices

JEE Main 2025 — 4 April, Evening Shift — Question 19

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]$ satisfy $A^{n}=A^{n-2}+A^{2}-l$ for $n \geq 3$ . Then the sum of all the elements of $A^{50}$ is:

Option A:
39
Option B:
52
Option C:
44
Option D:
53
Correct

Answer: D

Step-by-step solution

$A=\left[\begin{array}{lll}1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]$

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

$A^{3}=A+A^{2}-1$ $A^{3}=\left[\begin{array}{lll}1 & 0 & 0\\ 2 & 0 & 1\\ 1 & 1 & 0\end{array}\right]$

$A^{4}=A^{2}+A^{2}-I=2 A^{2}-I$

$A^{4}=\left[\begin{array}{lll}1 & 0 & 0\\ 2 & 1 & 0\\ 2 & 0 & 1\end{array}\right]$ and

$A^{5}=\left[\begin{array}{lll}1 & 0 & 0\\ 3 & 0 & 1\\ 2 & 1 & 0\end{array}\right]$

$A^{50}=\left[\begin{array}{ccc}1 & 0 & 0\\ 25 & 1 & 0\\ 25 & 0 & 1\end{array}\right]$

Sum of elements $=53$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 4 April, Evening Shift
Subject: Mathematics
Chapter: Matrices
Topic: Characteristic Equation & roots,application of cayley - hamilton theorem