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Si \( C=\left(\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right) \) ¿cuál de las siguientes matrices es la matriz D tal que \( \mathrm{C} \times \mathrm{D}=1 \) (matriz identidad)?

Ask by Jimenez Mathis. in Mexico
Mar 15,2025

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La matriz \( D \) que satisface \( C \times D = I \) es \[ D = \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix}. \]

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Para hallar la matriz \( D \) tal que \( C \times D = I \) (donde \( I \) es la matriz identidad), necesitamos encontrar la inversa de la matriz \( C \). La fórmula para la inversa de una matriz \( 2 \times 2 \) de la forma \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) es: \[ D = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \] Para la matriz \( C = \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} \): - \( a = 3 \), \( b = 2 \), \( c = 1 \), \( d = 1 \). - Determinante \( ad - bc = 3 \cdot 1 - 2 \cdot 1 = 1 \). Por lo tanto, la inversa es: \[ D = \frac{1}{1} \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix} \] Así que \( D = \begin{pmatrix} 1 & -2 \\ -1 & 3 \end{pmatrix} \) es la matriz que satisface la ecuación \( C \times D = I \).

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