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3. (6 puntos) Sean: \[ A=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right] \quad B=\left[\begin{array}{ccc}-2 & 3 & 0 \\ 1 & -4 & 1\end{array}\right] \quad C=\left[\begin{array}{cc}-1 & 1 \\ 0 & 3 \\ 2 & 4\end{array}\right] \] Calcule la matriz resultante, si existe, de la operación \( (3 A-2 B C)^{T} \) Pag. 1 de 1

Ask by Adkins Fowler. in Colombia
Mar 14,2025

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La matriz resultante de la operación \( (3A - 2BC)^T \) es: \[ \begin{pmatrix} -1 & -2 \\ -20 & 23 \end{pmatrix} \]

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Para calcular la matriz resultante de la operación \( (3 A - 2 B C)^{T} \), primero necesitamos calcular el producto \( B C \). La matriz \( B \) es de tamaño \( 2 \times 3 \) y la matriz \( C \) es de tamaño \( 3 \times 2 \). El producto \( B C \) dará como resultado una matriz de tamaño \( 2 \times 2 \). Calculando \( B C \): \[ B C = \begin{bmatrix}-2 & 3 & 0 \\ 1 & -4 & 1\end{bmatrix} \begin{bmatrix}-1 & 1 \\ 0 & 3 \\ 2 & 4\end{bmatrix} \] Calculemos el elemento \( (1,1) \): \[ (-2)(-1) + (3)(0) + (0)(2) = 2 \] Calculemos el elemento \( (1,2) \): \[ (-2)(1) + (3)(3) + (0)(4) = -2 + 9 = 7 \] Calculemos el elemento \( (2,1) \): \[ (1)(-1) + (-4)(0) + (1)(2) = -1 + 0 + 2 = 1 \] Calculemos el elemento \( (2,2) \): \[ (1)(1) + (-4)(3) + (1)(4) = 1 - 12 + 4 = -7 \] Por lo tanto, \( B C = \begin{bmatrix} 2 & 7 \\ 1 & -7 \end{bmatrix} \). Ahora, procedemos a calcular \( 2 B C \): \[ 2 B C = 2 \begin{bmatrix} 2 & 7 \\ 1 & -7 \end{bmatrix} = \begin{bmatrix} 4 & 14 \\ 2 & -14 \end{bmatrix} \] Siguiente, calculamos \( 3 A \): \[ 3 A = 3 \begin{bmatrix} 1 & -2 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 0 & 9 \end{bmatrix} \] Ahora realizamos la resta \( 3 A - 2 B C \): \[ 3 A - 2 B C = \begin{bmatrix} 3 & -6 \\ 0 & 9 \end{bmatrix} - \begin{bmatrix} 4 & 14 \\ 2 & -14 \end{bmatrix} = \begin{bmatrix} 3 - 4 & -6 - 14 \\ 0 - 2 & 9 - (-14) \end{bmatrix} = \begin{bmatrix} -1 & -20 \\ -2 & 23 \end{bmatrix} \] Finalmente, calculamos la transpuesta de la matriz resultante: \[ (3 A - 2 B C)^{T} = \begin{bmatrix} -1 & -2 \\ -20 & 23 \end{bmatrix} \] Por lo tanto, la matriz resultante de la operación \( (3 A - 2 B C)^{T} \) es: \[ \begin{bmatrix} -1 & -2 \\ -20 & 23 \end{bmatrix} \]

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