Considere a matriz \mathrm{A}=\left(\begin{array}{ccc}1 & 0 & 1 \\\\ 0 & 1 & 5 \\\\ 3 & 0 & 1\end{array}\right) e a matriz \mathrm{B}=\left(\begin{array}{lll}2 & 2 & 2 \\\\ 0 & 2 & 2 \\\\ 1 & 1 & 2\end{array}\right). Calcule:
\operatorname{det}\left(A^{4} B^{-2}\right).

Passo 1

O enunciado pede para calcular \det(A^4 B^{-2}). O conteúdo da matéria envolve as propriedades de determinantes:
\det(AB)=\det(A)\det(B)
\det(A^k)=(\det(A))^k
\det(B^{-1})=1/\det(B)

Passo 2

Calcular \det(A) por expansão na primeira linha:
\det(A) =1\cdot\bigl(1\cdot1-5\cdot0\bigr) -0\cdot\bigl(0\cdot1-5\cdot3\bigr) +1\cdot\bigl(0\cdot0-1\cdot3\bigr) =1+0-3=-2.

Passo 3

Calcular \det(B) por expansão na primeira linha:
\det(B) =2\cdot\bigl(2\cdot2-2\cdot1\bigr) -2\cdot\bigl(0\cdot2-2\cdot1\bigr) +2\cdot\bigl(0\cdot1-2\cdot1\bigr) =2\cdot2-2\cdot(-2)+2\cdot(-2) =4+4-4=4.

Passo 4

Usar as propriedades:
\det(A^4 B^{-2}) =\det(A^4)\,\det(B^{-2}) =(\det(A))^4\;(\det(B^{-1}))^2 =\frac{(\det(A))^4}{(\det(B))^2}.

Passo 5

Substituindo \det(A)=-2 e \det(B)=4:
\det(A^4 B^{-2}) =\frac{(-2)^4}{4^2} =\frac{16}{16} =1.

Resposta

1

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