Càlcul de la suma de potències d’una matriu

Donat que \( A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \), trobar \( A + A^2 + A^3 \).

Per resoldre el problema, hem de calcular \( A + A^2 + A^3 \), on \( A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \). Anem pas a pas:

Pas 1: Calcular \( A^2 \). Multipliquem la matriu \( A \) per ella mateixa:\[A^2 = A \cdot A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\]Calculem cada element de la matriu resultant:

• Element (1,1): \( 1 \cdot 1 + 0 \cdot 1 = 1 \)

• Element (1,2): \( 1 \cdot 0 + 0 \cdot 1 = 0 \)

• Element (2,1): \( 1 \cdot 1 + 1 \cdot 1 = 1 + 1 = 2 \)

• Element (2,2): \( 1 \cdot 0 + 1 \cdot 1 = 0 + 1 = 1 \)

Per tant:\[A^2 = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\]

Pas 2: Calcular \( A^3 \). Ara multipliquem \( A^2 \) per \( A \):\[A^3 = A^2 \cdot A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\]Calculem cada element:

• Element (1,1): \( 1 \cdot 1 + 0 \cdot 1 = 1 \)

• Element (1,2): \( 1 \cdot 0 + 0 \cdot 1 = 0 \)

• Element (2,1): \( 2 \cdot 1 + 1 \cdot 1 = 2 + 1 = 3 \)

• Element (2,2): \( 2 \cdot 0 + 1 \cdot 1 = 0 + 1 = 1 \)

Per tant:\[A^3 = \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}\]

Pas 3: Calcular \( A + A^2 + A^3 \). Sumem les matrius \( A \), \( A^2 \) i \( A^3 \):\[A + A^2 + A^3 = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}\]Sumem element per element:

• Element (1,1): \( 1 + 1 + 1 = 3 \)

• Element (1,2): \( 0 + 0 + 0 = 0 \)

• Element (2,1): \( 1 + 2 + 3 = 6 \)

• Element (2,2): \( 1 + 1 + 1 = 3 \)

Per tant:\[A + A^2 + A^3 = \begin{pmatrix} 3 & 0 \\ 6 & 3 \end{pmatrix}\]

Resposta final:\[\boxed{\begin{pmatrix} 3 & 0 \\ 6 & 3 \end{pmatrix}}\]