Puterile unei matrice din M_3(R)

[Filip Chindea]

Pentru orice \( n \in \mathbb{N}^{\ast} \), sa se calculeze \( A^n \), unde
\( A = \left( \begin{array}{ccc} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right) \in \mathcal{M}_3(\mathbb{R}) \).

[Mateescu Constantin]

Se stie, sau se arata usor prin inductie ca daca \( A\in\mathcal{M}_3(\mathbb{R}) \) exista sirurile \( (x_n)_{\small n\ge 1} \) , \( (y_n)_{\small n\ge 1} \) si \( (z_n)_{\small n\ge 1} \)

astfel incat : \( \fbox{\ A^n=x_n\cdot A^2\ +\ y_n\cdot A\ +\ z_n\cdot I_3\ } \) unde \( \left\{\ \begin{array}{llllll}
x_1=0 &\ ;\ & y_1=1 &\ ;\ & z_1=0 \\\\\\\\
x_2=1 &\ ;\ & y_2=0 &\ ;\ & z_2=0 \\\\\\\\
x_3=\tr\ (A) &\ ;\ & y_3=-\tr\ (A^{\ast}) &\ ;\ & z_3=\det\ A \\\\\\\\
x_{n+1}=x_3\cdot x_n+y_n &\ ;\ & y_{n+1}=y_3\cdot x_n+z_n &\ ;\ & z_{n+1}=z_3\cdot x_n\end{array} \)

In cazul nostru \( x_3=\tr(A)=7 \) , \( y_3=\tr(A^{\ast})=0 \) si \( z_3=\det A=-36 \) . Prin urmare avem \( x_{n+2}=7x_{n+1}-36x_{n-1} \) ,

sir cu ecuatia caracteristica \( r^3=7r^2-36 \) (care-i totuna cu ecuatia caracteristica a matricei \( A \)) . Asadar, sirul \( (x_n)_{\small n\ge 1} \) este de forma :

\( x_n=\alpha\cdot (-2)^n+\beta\cdot 3^n+\gamma\cdot 6^n \) . Cunoscand primii termeni ai sirului se obtine usor ca : \( \fbox{\ \begin{array}x_n=\frac 1{40} & \cdot & (-2)^n & - & \frac 1{15} & \cdot & 3^n & + & \frac 1{24} & \cdot & 6^n\end{array}\ } \) .

Imediat gasim : \( \fbox{\ \begin{array}y_n=-\frac 9{10} & \cdot & (-2)^{n-2} & + & \frac {12}5 & \cdot & 3^{n-2} & - & \frac 32 & \cdot & 6^{n-2}\end{array}\ } \) precum si \( \fbox{\ \begin{array}z_n=-\frac 9{10} & \cdot & (-2)^{n-1}+\frac {12}5 & \cdot & 3^{n-1}-\frac 32 & \cdot & 6^{n-1}\end{array}\ } \) .

In fine, \( \begin{array}{ccc} A^n=x_n & \cdot & \left\(\begin{array}{ccc} 11 & 9 & 7 \\ \\ 9 & 27 & 9 \\ \\ 7 & 9 & 11\end{array}\right\) & + & y_n & \cdot & \left\(\begin{array}{ccc} 1 & 1 & 3 \\ \\ 1 & 5 & 1 \\ \\ 3 & 1 & 1\end{array}\right\) & + & z_n & \cdot & \left\(\begin{array}{ccc} 1 & 0 & 0 \\ \\ 0 & 1 & 0 \\ \\ 0 & 0 & 1\end{array}\right\)=\left\(\begin{array}{ccccc} 11x_n+y_n+z_n & 9x_n+y_n & 7x_n+3y_n \\\\\\\\ 9x_n+y_n & 27x_n+5y_n+z_n & 9x_n+y_n \\\\\\\\ 7x_n+3y_n & 9x_n+y_n & 11x_n+y_n+z_n\end{array}\right\)\end{array} \)

unde sirurile \( (x_n)_{\small n\ge 1} \) , \( (y_n)_{\small n\ge 1} \) si \( (z_n)_{\small n\ge 1} \) au fost determinate mai sus .