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QJEEM24Mab172: JEE Main 2024 Mathematics - Class 12 JEEM24Mab172 Solution: Let . If the sum of the diagonal elements of is , then is equal …

Let A=[2amp;−11amp;1]A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right] . If the sum of the diagonal elements of A13A^{13} is 3n3^n , then nn is equal to ________.

Step-by-Step Verified Solution

amp;A=[2amp;−11amp;1]amp;A2=[2amp;−11amp;1][2amp;−11amp;1]amp;A2=[3amp;−33amp;0]=3[1amp;−11amp;0]amp;A4=9[0amp;−11amp;−1]amp;A8=81[−1amp;1−1amp;0]amp;A12=729[1amp;00amp;1]amp;A13=729[1amp;00amp;1][2amp;−11amp;1]amp;A13=[1458amp;−729729amp;729]\begin{aligned}& A=\left[\begin{array}{cc}2 & -1 \\1 & 1\end{array}\right] \\& A^2=\left[\begin{array}{cc}2 & -1 \\1 & 1\end{array}\right]\left[\begin{array}{cc}2 & -1 \\1 & 1\end{array}\right] \\& A^2=\left[\begin{array}{cc}3 & -3 \\3 & 0\end{array}\right]=3\left[\begin{array}{cc}1 & -1 \\1 & 0\end{array}\right] \\& A^4=9\left[\begin{array}{ll}0 & -1 \\1 & -1\end{array}\right] \\& A^8=81\left[\begin{array}{ll}-1 & 1 \\-1 & 0\end{array}\right] \\& A^{12}=729\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right] \\& A^{13}=729\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]\left[\begin{array}{cc}2 & -1 \\1 & 1\end{array}\right] \\& A^{13}=\left[\begin{array}{cc}1458 & -729 \\729 & 729\end{array}\right]\end{aligned}

amp; Sum =2187=3namp;37=3namp;n=7\begin{aligned}& \text { Sum }=2187=3^n \\& 3^7=3^n \\& n=7\end{aligned}

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