Whittaker Powell
08/12/2024 · Junior High School
15 Dadas \( A \mathrm{e} B \), acha \( X \) tal que \( 2 X-B^{2}=A \cdot B \) \[ A=\left(\begin{array}{lll}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right) \quad B=\left(\begin{array}{ccc}1 & 0 & -1 \\ 1 & 1 & 1 \\ 0 & 0 & 1\end{array}\right) \]
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Para resolver a equação \( 2X - B^2 = A \cdot B \), precisamos primeiro calcular \( B^2 \) e \( A \cdot B \).
### Passo 1: Calcular \( B^2 \)
A matriz \( B \) é dada por:
\[
B = \begin{pmatrix}
1 & 0 & -1 \\
1 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix}
\]
Calculamos \( B^2 = B \cdot B \):
\[
B^2 = \begin{pmatrix}
1 & 0 & -1 \\
1 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0 & -1 \\
1 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix}
\]
Calculando cada elemento da matriz resultante:
- Para a primeira linha:
- \( (1 \cdot 1 + 0 \cdot 1 + (-1) \cdot 0) = 1 \)
- \( (1 \cdot 0 + 0 \cdot 1 + (-1) \cdot 0) = 0 \)
- \( (1 \cdot -1 + 0 \cdot 1 + (-1) \cdot 1) = -1 - 1 = -2 \)
- Para a segunda linha:
- \( (1 \cdot 1 + 1 \cdot 1 + 1 \cdot 0) = 1 + 1 = 2 \)
- \( (1 \cdot 0 + 1 \cdot 1 + 1 \cdot 0) = 0 + 1 = 1 \)
- \( (1 \cdot -1 + 1 \cdot 1 + 1 \cdot 1) = -1 + 1 + 1 = 1 \)
- Para a terceira linha:
- \( (0 \cdot 1 + 0 \cdot 1 + 1 \cdot 0) = 0 \)
- \( (0 \cdot 0 + 0 \cdot 1 + 1 \cdot 0) = 0 \)
- \( (0 \cdot -1 + 0 \cdot 1 + 1 \cdot 1) = 1 \)
Assim, temos:
\[
B^2 = \begin{pmatrix}
1 & 0 & -2 \\
2 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix}
\]
### Passo 2: Calcular \( A \cdot B \)
A matriz \( A \) é dada por:
\[
A = \begin{pmatrix}
1 & 0 & 1 \\
1 & 1 & 0 \\
0 & 0 & 2
\end{pmatrix}
\]
Calculamos \( A \cdot B \):
\[
A \cdot B = \begin{pmatrix}
1 & 0 & 1 \\
1 & 1 & 0 \\
0 & 0 & 2
\end{pmatrix} \cdot \begin{pmatrix}
1 & 0 & -1 \\
1 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix}
\]
Calculando cada elemento da matriz resultante:
- Para a primeira linha:
- \( (1 \cdot 1 + 0 \cdot 1 + 1 \cdot 0) = 1 \)
- \( (1 \cdot 0 + 0 \cdot 1 + 1 \cdot 0) = 0 \)
- \( (1 \cdot -1 + 0 \cdot 1 + 1 \cdot 1) = -1 + 1 = 0 \)
- Para a segunda linha:
- \( (1 \cdot 1 + 1 \cdot 1 + 0 \cdot 0) = 1 + 1 = 2 \)
- \( (1 \cdot 0 + 1 \cdot 1 + 0 \cdot 0) = 0 + 1 = 1 \)
- \( (1 \cdot -1 + 1 \cdot 1 + 0 \cdot 1) = -
Quick Answer
Para resolver a equação \( 2X - B^2 = A \cdot B \), primeiramente calcule \( B^2 \) e \( A \cdot B \). Após calcular essas matrizes, encontre \( X \) para que a equação seja satisfeita.
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