Let A=[ array cc 2 & -1 1 & 1 array ] . If the sum of the diagonal elements of A^ 13 is 3^n , then n is equal to_________

Correct answer: 0 A= [ array cc 2 & -1 1 & 1 array ] aligned & A^2=[ array cc 2 & -1 1 & 1 array ][ array cc 2 & -1 1 & 1 array ]=[ array cc 3 & -3 3 & 0 array ] & A^3=[ array cc 3 & -3 3 & 0 array ][ array cc 2 & -1 1 & 1 array ]=[ array cc 3 & -6 6 & -3 array ] & A^4=[ array cc 3 & -6 6 & -3 array ][ array cc 2 & -1 1 & 1 array ]=[ array cc 0 & -9 9 & -9 array ] & A^5=[ array cc 0 & -9 9 & -9 array ][ array cc 2 & -1 1 & 1 array ]=[ array cc -9 & -9 9 & -18 array ] & A^6=[ array cc -9 & -9 9 & -18 array ][ array cc 2 & -1 1 & 1 array ]=[ array cc -27 & 0 0 & -27 array ] & A^7=[ array cc -27 & -0 0 & -27 array ][ array cc -54 & 27 -27 & -27 array ]=[ array cc 3^6 × 2 & -27^2 27^2 & 3^6 array ] & 3^7=3^n n=7 aligned

JEE Main 20248 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 8 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Matrices

Let A=[ array cc 2 & -1 1 & 1 array ] . If the sum of the diagonal...

Let

A=\left[

\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}

\right]

. If the sum of the diagonal elements of

A^{13}

3^n

, then

n

is equal to_________

Correct answer

Step-by-step solution

\left[

\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}

\right]

\begin{aligned} & A^2=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -3 \\ 3 & 0\end{array}\right] \\ & A^3=\left[\begin{array}{cc}3 & -3 \\ 3 & 0\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -6 \\ 6 & -3\end{array}\right] \\ & A^4=\left[\begin{array}{cc}3 & -6 \\ 6 & -3\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}0 & -9 \\ 9 & -9\end{array}\right] \\ & A^5=\left[\begin{array}{cc}0 & -9 \\ 9 & -9\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}-9 & -9 \\ 9 & -18\end{array}\right] \\ & A^6=\left[\begin{array}{cc}-9 & -9 \\ 9 & -18\end{array}\right]\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}-27 & 0 \\ 0 & -27\end{array}\right] \\ & A^7=\left[\begin{array}{cc}-27 & -0 \\ 0 & -27\end{array}\right]\left[\begin{array}{cc}-54 & 27 \\ -27 & -27\end{array}\right]=\left[\begin{array}{cc}3^6 \times 2 & -27^2 \\ 27^2 & 3^6\end{array}\right] \\ & 3^7=3^n \Rightarrow n=7\end{aligned}

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