Question

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

Accepted Answer

(A) There exists a $2 \times 2$ invertible matrix N with real entries such that
$$
\mathrm{MN}=\mathrm{N}\left[\begin{array}{ll}
1 & 1 \
0 & 1
\end{array}\right]
$$; (C) For any two given integers m and n , there exist unique integers x and y such that
$$
px+qy=m \text { and } rx+sy=n
$$; (D) For each positive real number t , the system of linear equations
$$
\begin{gathered}
(a+t) x+b y=1 \
cx+(d+t) y=-1
\end{gathered}
$$
has a unique solution

Answer

(B) The value of a is 378

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Matrices

Solution

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