Seja A=\left(\begin{array}{cc}1 & 2 \\\\ 4 & -3\end{array}\right). Então:
(a) Encontre A^{2} e A^{3}.
(b) Determine \mathrm{f}(\mathrm{A}) onde f(x)=2 x^{3}-4 x+5.
(c) Mostre que A anula o polinômio g(x)=x^{2}+2 x-11.

Passo 1

O enunciado dá a matriz
A = \begin{pmatrix}1 & 2 \\\\ 4 & -3\end{pmatrix}
e pede:
(a) calcular A^2 e A^3;
(b) avaliar o polinômio f(x)=2x^3-4x+5 em A, ou seja, f(A);
(c) mostrar que A anula g(x)=x^2+2x-11, isto é, que g(A)=0.

Passo 2

Para calcular A^2, usamos multiplicação de matrizes:
A^2 = \begin{pmatrix}1 & 2 \\\\ 4 & -3\end{pmatrix} \begin{pmatrix}1 & 2 \\\\ 4 & -3\end{pmatrix} = \begin{pmatrix} 1\cdot1 + 2\cdot4 & 1\cdot2 + 2\cdot(-3) \\\\ 4\cdot1 + (-3)\cdot4 & 4\cdot2 + (-3)\cdot(-3) \end{pmatrix} = \begin{pmatrix}9 & -4 \\\\ -8 & 17\end{pmatrix}.

Passo 3

Para obter A^3, multiplicamos A^2 por A:
A^3 = A^2\,A = \begin{pmatrix}9 & -4 \\\\ -8 & 17\end{pmatrix} \begin{pmatrix}1 & 2 \\\\ 4 & -3\end{pmatrix} = \begin{pmatrix} 9\cdot1 + (-4)\cdot4 & 9\cdot2 + (-4)\cdot(-3) \\\\ -8\cdot1 + 17\cdot4 & -8\cdot2 + 17\cdot(-3) \end{pmatrix} = \begin{pmatrix}-7 & 30 \\\\ 60 & -67\end{pmatrix}.

Passo 4

Para avaliar f(A) com f(x)=2x^3-4x+5, usamos
f(A) = 2A^3 - 4A + 5I.
Já temos
2A^3 = \begin{pmatrix}-14 & 60 \\\\ 120 & -134\end{pmatrix},\quad -4A = \begin{pmatrix}-4 & -8 \\\\ -16 & 12\end{pmatrix},\quad 5I = \begin{pmatrix}5 & 0 \\\\ 0 & 5\end{pmatrix}.
Somando,
f(A) = \begin{pmatrix}-14 & 60 \\\\ 120 & -134\end{pmatrix} + \begin{pmatrix}-4 & -8 \\\\ -16 & 12\end{pmatrix} + \begin{pmatrix}5 & 0 \\\\ 0 & 5\end{pmatrix} = \begin{pmatrix}-13 & 52 \\\\ 104 & -117\end{pmatrix}.

Passo 5

Para mostrar que g(A)=A^2+2A-11I=0, note que
A^2+2A-11I = \begin{pmatrix}9 & -4 \\\\ -8 & 17\end{pmatrix} + 2\begin{pmatrix}1 & 2 \\\\ 4 & -3\end{pmatrix} - 11\begin{pmatrix}1 & 0 \\\\ 0 & 1\end{pmatrix} = \begin{pmatrix}0 & 0 \\\\ 0 & 0\end{pmatrix}.
Logo, g(A)=0.

Resposta

(a)
A^2 = \begin{pmatrix}9 & -4 \\\\ -8 & 17\end{pmatrix},\quad A^3 = \begin{pmatrix}-7 & 30 \\\\ 60 & -67\end{pmatrix}.
(b)
f(A) = \begin{pmatrix}-13 & 52 \\\\ 104 & -117\end{pmatrix}.
(c)
g(A)=A^2+2A-11I=0.

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