JEE Advanced 2026 · Paper 1 · Mathematics Q8

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MathematicsMultiple correct+4 / -1Question 8 of 48

Consider the matrix
$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$
Let $p,q,r,s,a,b,c$ and $d$ be integers such that
$M^{26}=\begin{bmatrix}p&q\\r&s\end{bmatrix}\quad\text{and}\quad\sum_{k=1}^{26}M^k=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$
Then which of the following statements is (are) TRUE ?

There exists a

2\times2

invertible matrix

N

with real entries such that

MN=N\begin{bmatrix}1&1\\0&1\end{bmatrix}

The value of

a

is 378

For any two given integers

m

and

n

, there exist unique integers

x

and

y

such that

px+qy=m

and

rx+sy=n

For each positive real number

t

, the system of linear equations

(a+t)x+by=1

cx+(d+t)y=-1

has a unique solution

Answer and worked solution

Official answer: A, C, D

Structure of $M$ . Write $M=I+N$ with $N=\begin{bmatrix}1&-1\\1&-1\end{bmatrix}$ . Then $N^2=0$ , so
$M^k=I+kN=\begin{bmatrix}1+k&-k\\k&1-k\end{bmatrix}.$

(A) $M$ has the single eigenvalue 1 (characteristic polynomial $(\lambda-1)^2$ ) and $M\ne I$ , so it is similar to the Jordan block $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ . True.

(B) $\sum_{k=1}^{26}M^k=26I+\left(\sum k\right)N=26I+351N$ , so $a=26+351=377$ . False.

(C) $M^{26}=\begin{bmatrix}27&-26\\26&-25\end{bmatrix}$ has determinant $1$ , so its inverse has integer entries and the system has exactly one integer solution for any integers $m,n$ . True.

(D) The coefficient matrix is $(26+t)I+351N$ . Since $N$ is nilpotent, its determinant is $(26+t)^2>0$ for $t>0$ . True.

Topics:Matrix algebra Systems of linear equations Determinants and properties

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JEE Advanced 2026 · Paper 1, Mathematics question 8