QUESTION PAPER

JEE Advanced 2026 · Paper 1 · Mathematics · Q.8 · Multiple-select

Consider the matrix

M = \begin{pmatrix}2 & -1\\1 & 0\end{pmatrix}

Let $p$ , $q$ , $r$ , $s$ , $a$ , $b$ , $c$

M^{26} = \begin{pmatrix}p & q\\r & s\end{pmatrix}

\sum_{k=1}^{26} M^k = \begin{pmatrix}a & b\\c & d\end{pmatrix}

Then which of the following statements is (are) TRUE ?

A
There exists a $2 \times 2$ invertible matrix $N$ with real entries such that
$MN = N\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$

QUESTION CONTEXTDocumented fact

Question and context

JEE Advanced 2026 Paper 1, Mathematics, Question 8 (multiple-select), printed page 5. The question gives

M = \begin{pmatrix}2 & -1\\1 & 0\end{pmatrix}

RESEARCH NOTE 0112 SECTIONS

Decoding a Matrix with a Repeated Eigenvalue

A structural reading of JEE Advanced 2026 Paper 1 Mathematics Question 8, showing how a nilpotent shift simplifies powers, sums, and related properties.

02Documented fact

Official answer key

The official final solutions list A, C, and D as the correct options. The note below develops a structural explanation for those claims and for why B is false.

SourceOfficial JEE Advanced 2026 Final Solutions, Paper 1, Q.8

03Analysis

The first clue: repeated eigenvalue

The characteristic polynomial is

\det(M - \lambda I) = (2 - \lambda)(-\lambda) + 1 = \lambda^2 - 2\lambda + 1 = (\lambda - 1)^2

Question and context

Decoding a Matrix with a Repeated Eigenvalue

Official answer key

The first clue: repeated eigenvalue

Decode the matrix

Why the structure makes high powers easy

the Jordan connection

summing the powers

why determinant 1 matters

the same structure strikes again

The big picture

Beyond this question: a high-powers toolbox

Final takeaway