Dadas as matrizes A=\left[\begin{array}{ccc}1 & 2 & 3 \\\\ 0 & -1 & 4 \\\\ 5 & 6 & 1\end{array}\right] \quad B=\left[\begin{array}{ccc}-1 & 5 & 7 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] \quad C=\left[\begin{array}{ccc}5 & -7 & 4 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] Calcule: -A+4 B-1 / 3 C Passo 1Temos as matrizes A = \begin{pmatrix} 1 & 2 & 3\\\\ 0 & -1 & 4\\\\ 5 & 6 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 5 & 7\\\\ 2 & 1 & -3\\\\ 0 & 4 & 6 \end{pmatrix}, \quad C = \begin{pmatrix} 5 & -7 & 4\\\\ 2 & 1 & -3\\\\ 0 & 4 & 6 \end{pmatrix}. Queremos calcular a combinação linear -A + 4B - \tfrac{1}{3}C. Vamos usar operações de multiplicação escalar e soma de matrizes componente a componente. Passo 2Calcular 4B, multiplicando cada entrada de B por 4: 4B = \begin{pmatrix} 4\cdot(-1) & 4\cdot5 & 4\cdot7\\\\ 4\cdot2 & 4\cdot1 & 4\cdot(-3)\\\\ 4\cdot0 & 4\cdot4 & 4\cdot6 \end{pmatrix} = \begin{pmatrix} -4 & 20 & 28\\\\ 8 & 4 & -12\\\\ 0 & 16 & 24 \end{pmatrix}. Passo 3Calcular \tfrac{1}{3}C, multiplicando cada entrada de C por \tfrac{1}{3}: \tfrac{1}{3}C = \begin{pmatrix} \tfrac{5}{3} & -\tfrac{7}{3} & \tfrac{4}{3}\\\ \tfrac{2}{3} & \tfrac{1}{3} & -1\\\\ 0 & \tfrac{4}{3} & 2 \end{pmatrix}. Passo 4Somar -A e 4B. Primeiro, -A = \begin{pmatrix} -1 & -2 & -3\\\\ 0 & 1 & -4\\\\ -5 & -6 & -1 \end{pmatrix}. Então -A + 4B = \begin{pmatrix} -1 & -2 & -3\\\\ 0 & 1 & -4\\\\ -5 & -6 & -1 \end{pmatrix} + \begin{pmatrix} -4 & 20 & 28\\\\ 8 & 4 & -12\\\\ 0 & 16 & 24 \end{pmatrix} = \begin{pmatrix} -5 & 18 & 25\\\\ 8 & 5 & -16\\\\ -5 & 10 & 23 \end{pmatrix}. Passo 5Agora subtrair \tfrac{1}{3}C do resultado do passo anterior, componente a componente: \begin{pmatrix} -5 & 18 & 25\\\\ 8 & 5 & -16\\\\ -5 & 10 & 23 \end{pmatrix} - \begin{pmatrix} \tfrac{5}{3} & -\tfrac{7}{3} & \tfrac{4}{3}\\\ \tfrac{2}{3} & \tfrac{1}{3} & -1\\\\ 0 & \tfrac{4}{3} & 2 \end{pmatrix} = \begin{pmatrix} -\tfrac{20}{3} & \tfrac{61}{3} & \tfrac{71}{3}\\\ \tfrac{22}{3} & \tfrac{14}{3} & -15\\\\ -5 & \tfrac{26}{3} & 21 \end{pmatrix}. Resposta -\,A \;+\;4\,B\;-\;\tfrac{1}{3}C = \begin{pmatrix} -\tfrac{20}{3} & \tfrac{61}{3} & \tfrac{71}{3}\\\ \tfrac{22}{3} & \tfrac{14}{3} & -15\\\\ -5 & \tfrac{26}{3} & 21 \end{pmatrix}. MOSTRAR SOLUÇÃO COMPLETAExercícios relacionadosDadas as matrizes A=\left[\begin{array}{ccc}1 & 2 & 3 \\\\ 0 & -1 & 4 \\\\ 5 & 6 & 1\end{array}\right] \quad B=\left[\begin{array}{ccc}-1 & 5 & 7 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] \quad C=\left[\begin{array}{ccc}5 & -7 & 4 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] Calcule: A+B-2 C Dadas as matrizes A=\left[\begin{array}{ccc}1 & 2 & 3 \\\\ 0 & -1 & 4 \\\\ 5 & 6 & 1\end{array}\right] \quad B=\left[\begin{array}{ccc}-1 & 5 & 7 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] \quad C=\left[\begin{array}{ccc}5 & -7 & 4 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] Calcule: 2 / 5(\mathrm{~B}-\mathrm{C}) Dadas as matrizes A=\left[\begin{array}{ccc}1 & 2 & 3 \\\\ 0 & -1 & 4 \\\\ 5 & 6 & 1\end{array}\right] \quad B=\left[\begin{array}{ccc}-1 & 5 & 7 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] \quad C=\left[\begin{array}{ccc}5 & -7 & 4 \\\\ 2 & 1 & -3 \\\\ 0 & 4 & 6\end{array}\right] Calcule: \frac{B-C}{2}-4 \mathrm{~A} Dadas as matrizes A=\left[\begin{array}{ccc}0 & 1 & 3 \\\\ 4 & -2 & 5 \\\\ -1 & 2 & 7\end{array}\right], \quad B=\left[\begin{array}{ccc}2 & 4 & -1 \\\\ 3 & 0 & 1 \\\\ 1 & 2 & 5\end{array}\right] e C=\left[\begin{array}{ccc}-1 & 2 & 0 \\\\ 3 & 4 & 6 \\\\ 1 & 0 & -1\end{array}\right] Obtenha AB