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 1

Temos 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 2

Calcular 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 3

Calcular \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 4

Somar -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 5

Agora 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}.

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