Barber Ryan
07/06/2024 · Junior High School
2.- Realiza las operaciones con matrices si \( A=\begin{array}{llcl}2 & 5 \\ 7 & 8\end{array} B=\begin{array}{cc}23 & 7 \\ 1 & 0\end{array} \) \& \( C=\begin{array}{cc}2 & 9 \\ 4 & 10\end{array} \) a) \( C^{2} \) b) \( A+C \), c) \( (A+C)^{2,} \) d) \( B+C^{2,} \), e) \( A * B+B^{*} C+C^{*} B+B^{*} A \) f) \( B^{2}+B^{3,} \) g) \( 3 C \), h) \( 5(A+B+C) \)
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Step-by-step Solution
Claro, aquí tienes las soluciones para las operaciones con matrices:
a) \( C^{2} \)
Para calcular \( C^{2} \), multiplicamos la matriz \( C \) por sí misma:
\[ C = \begin{array}{cc}2 & 9 \\ 4 & 10\end{array} \]
\[ C^{2} = C \cdot C = \begin{array}{cc}2 & 9 \\ 4 & 10\end{array} \cdot \begin{array}{cc}2 & 9 \\ 4 & 10\end{array} \]
\[ C^{2} = \begin{array}{cc}2 \cdot 2 + 9 \cdot 4 & 2 \cdot 9 + 9 \cdot 10 \\ 4 \cdot 2 + 10 \cdot 4 & 4 \cdot 9 + 10 \cdot 10\end{array} \]
\[ C^{2} = \begin{array}{cc}4 + 36 & 18 + 90 \\ 8 + 40 & 36 + 100\end{array} \]
\[ C^{2} = \begin{array}{cc}40 & 108 \\ 48 & 136\end{array} \]
b) \( A + C \)
Para sumar matrices \( A \) y \( C \), deben tener el mismo tamaño:
\[ A = \begin{array}{cc}2 & 5 \\ 7 & 8\end{array}, \quad C = \begin{array}{cc}2 & 9 \\ 4 & 10\end{array} \]
\[ A + C = \begin{array}{cc}2 + 2 & 5 + 9 \\ 7 + 4 & 8 + 10\end{array} \]
\[ A + C = \begin{array}{cc}4 & 14 \\ 11 & 18\end{array} \]
c) \( (A + C)^{2} \)
Primero, calculamos \( A + C \) y luego multiplicamos el resultado por sí mismo:
\[ A + C = \begin{array}{cc}4 & 14 \\ 11 & 18\end{array} \]
\[ (A + C)^{2} = (A + C) \cdot (A + C) \]
\[ (A + C)^{2} = \begin{array}{cc}4 & 14 \\ 11 & 18\end{array} \cdot \begin{array}{cc}4 & 14 \\ 11 & 18\end{array} \]
\[ (A + C)^{2} = \begin{array}{cc}4 \cdot 4 + 14 \cdot 11 & 4 \cdot 14 + 14 \cdot 18 \\ 11 \cdot 4 + 18 \cdot 11 & 11 \cdot 14 + 18 \cdot 18\end{array} \]
\[ (A + C)^{2} = \begin{array}{cc}16 + 154 & 56 + 252 \\ 44 + 198 & 154 + 324\end{array} \]
\[ (A + C)^{2} = \begin{array}{cc}170 & 308 \\ 242 & 478\end{array} \]
d) \( B + C^{2} \)
Primero, calculamos \( C^{2} \) y luego sumamos la matriz \( B \) al resultado:
\[ B = \begin{array}{cc}23 & 7 \\ 1 & 0\end{array}, \quad C^{2} = \begin{array}{cc}40 & 108 \\ 48 & 136\end{array} \]
\[ B + C^{2} = \begin{array}{cc}23 & 7 \\ 1 & 0\end{array} + \begin{array}{cc}40 & 108 \\ 48 & 136\end{array} \]
\[ B + C^{2} = \begin{array}{cc}23 + 40 & 7 + 108 \\ 1 + 48 & 0 + 136\end{array} \]
\[ B + C^{2} = \begin{array}{cc}63 & 115 \\ 49 & 136\end{array} \]
e) \( A * B + B^{*} C + C^{*} B + B^{*} A \)
Para esta operación, primero calculamos cada multiplicación de matrices y luego las sumamos:
\[ A \cdot B = \begin{array}{cc}2 & 5 \\ 7 & 8\end
Quick Answer
a) \( C^{2} = \begin{array}{cc}40 & 108 \\ 48 & 136\end{array} \)
b) \( A + C = \begin{array}{cc}4 & 14 \\ 11 & 18\end{array} \)
c) \( (A + C)^{2} = \begin{array}{cc}170 & 308 \\ 242 & 478\end{array} \)
d) \( B + C^{2} = \begin{array}{cc}63 & 115 \\ 49 & 136\end{array} \)
e) \( A * B + B^{*} C + C^{*} B + B^{*} A \) (calcular cada multiplicación y sumar)
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