(Solved):   Na matrix \( P \) which transform the following matrices to diagonal form. Hence calculate ...

 

Na matrix \( P \) which transform the following matrices to diagonal form. Hence calculate the power matrix. If \( A=\left[\begin{array}{lll}1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1\end{array}\right] \) calculate \( A^{4} \). If \( A=\left[\begin{array}{rrr}3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3\end{array}\right] \) calculate \( A^{4} \) Ans. \( \left[\begin{array}{rrr}251 & 485 & 235 \\ 485 & 1051 & 485 \\ 235 & 485 & 251\end{array}\right] \) If \( A=\left[\begin{array}{rrr}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right] \), calculate \( A^{6} \) Aass. \( \left[\begin{array}{rrr}251 & -405 & 235 \\ -405 & 891 & -405 \\ 235 & -405 & 251\end{array}\right] \) If \( A=\left[\begin{array}{rrr}1 & 1 & 1 \\ 0 & 2 & 1 \\ -4 & -4 & 3\end{array}\right] \) calculate \( A^{3} \) Ans. \( \left[\begin{array}{rrr}1446 & -1365 & 1365 \\ -1365 & 1446 & -1365 \\ 1365 & -1365 & 1446\end{array}\right] \) Ans. \( \left[\begin{array}{lll}-[12099 & 12355 & 6305 \\ -12100 & 12356 & 6305 \\ -13120 & 13120 & 6561\end{array}\right] \) If \( A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right] \) then show that \( A^{n}=A^{n-2}+A^{2}-I \), for \( n \geq 3 \). Hence find \( A^{50} \).