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Dada la siguiente matriz \( A \) \( A=\left(\begin{array}{ll}3 & 4 \\ 2 & 5\end{array}\right) \) ¿Cuál es la inversa de la matriz A? Nota: \( A^{-1}=\frac{1}{\operatorname{det}(A)} \cdot\left(\begin{array}{cc}d & -b \\ -c & a\end{array}\right) \)

Ask by Fernandez King. in Mexico
Mar 10,2025

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Answer

La inversa de la matriz \( A \) es: \[ A^{-1} = \begin{pmatrix} \frac{5}{7} & -\frac{4}{7} \\ -\frac{2}{7} & \frac{3}{7} \end{pmatrix} \]

Solution

1. Identificamos la matriz \( A \): \[ A = \begin{pmatrix} 3 & 4 \\ 2 & 5 \end{pmatrix} \] 2. Se calcula el determinante de \( A \): \[ \operatorname{det}(A) = (3)(5) - (4)(2) = 15 - 8 = 7 \] 3. Se utiliza la fórmula para la inversa de una matriz \( 2 \times 2 \): \[ A^{-1} = \frac{1}{\operatorname{det}(A)} \cdot \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \] donde en \( A \) tenemos \( a = 3 \), \( b = 4 \), \( c = 2 \) y \( d = 5 \). 4. Sustituimos los valores en la fórmula: \[ A^{-1} = \frac{1}{7} \cdot \begin{pmatrix} 5 & -4 \\ -2 & 3 \end{pmatrix} \] 5. La inversa de la matriz \( A \) es: \[ A^{-1} = \begin{pmatrix} \frac{5}{7} & -\frac{4}{7} \\ -\frac{2}{7} & \frac{3}{7} \end{pmatrix} \]

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Para encontrar la inversa de la matriz \( A \), primero debes calcular el determinante de \( A \). Esto se hace con la fórmula \( \operatorname{det}(A) = ad - bc \), donde \( a = 3, b = 4, c = 2, d = 5 \). Así que el determinante es: \[ \operatorname{det}(A) = (3)(5) - (4)(2) = 15 - 8 = 7. \] Luego, utilizando la fórmula de la inversa: \[ A^{-1} = \frac{1}{7} \cdot \left(\begin{array}{cc}5 & -4 \\ -2 & 3\end{array}\right) = \left(\begin{array}{cc}\frac{5}{7} & -\frac{4}{7} \\ -\frac{2}{7} & \frac{3}{7}\end{array}\right). \] Por lo tanto, la inversa de la matriz \( A \) es \[ A^{-1} = \left(\begin{array}{cc}\frac{5}{7} & -\frac{4}{7} \\ -\frac{2}{7} & \frac{3}{7}\end{array}\right). \] ¡Y así obtienes la inversa de la matriz A!

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