Exercício 7 Sejam as matrizes A=\left[\begin{array}{cc}2 & 1 \\\\ 0 & -1 \\\\ 3 & 2\end{array}\right] \quad B=\left[\begin{array}{cc}0 & 1 \\\\ 7 & 3 \\\\ -4 & 5\end{array}\right] \quad C=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ 7 & 3 & 1 \\\\ -4 & 5 & \frac{1}{2}\end{array}\right] \quad D= \left[\begin{array}{ccc}3 & 2 & 1 \\\\ -4 & 6 & 5\end{array}\right] \quad E=\left[\begin{array}{ll}0 & 1 \\\\ 7 & 3\end{array}\right] \quad F=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ -1 & 3 & -2 \\\\ 4 & 4 & -4\end{array}\right] \quad G=\left[\begin{array}{cc}-\frac{3}{7} & \frac{1}{7} \\\\ 1 & 0\end{array}\right] \quad H=\left[\begin{array}{c}4 \\\\ -19 \\\\ 26\end{array}\right] Calcule os seguintes produtos de matrizes A \cdot B^{T}-2 C Passo 1O enunciado dá as matrizes A = \begin{pmatrix}2 & 1 \\[6pt] 0 & -1 \\[6pt] 3 & 2\end{pmatrix},\quad B = \begin{pmatrix}0 & 1 \\[3pt] 7 & 3 \\[3pt] -4 & 5\end{pmatrix},\quad C = \begin{pmatrix}0 & 1 & 0 \\[3pt] 7 & 3 & 1 \\[3pt] -4 & 5 & \tfrac12\end{pmatrix} e pede o produto \;A\cdot B^T - 2\,C. Vamos usar as operações de transposta, multiplicação de matrizes, multiplicação por escalar e subtração de matrizes.] Passo 2Primeiro calculamos a transposta de B. Como B é 3\times2, B^T = \begin{pmatrix}0 & 7 & -4 \\[3pt] 1 & 3 & 5\end{pmatrix}. Agora multiplicamos A (de ordem 3\times2) por B^T (de ordem 2\times3). Pela definição (A\,B^T)_{ij} \;=\;\sum_{k=1}^{2}A_{ik}\,(B^T)_{kj}. Em particular, (A\,B^T)_{11} = 2\cdot0 + 1\cdot1 = 1,\quad (A\,B^T)_{12} = 2\cdot7 + 1\cdot3 = 17,\quad (A\,B^T)_{13} = 2\cdot(-4) + 1\cdot5 = -3. Repetindo o cálculo para todas as linhas, obtemos A\,B^T = \begin{pmatrix} 1 & 17 & -3 \\[3pt] -1 & -3 & -5 \\[3pt] 2 & 27 & -2 \end{pmatrix}.] Passo 3Multiplicamos C por 2, escala a cada entrada: 2\,C = \begin{pmatrix} 0 & 2 & 0 \\[3pt] 14 & 6 & 2 \\[3pt] -8 & 10 & 1 \end{pmatrix}.] Passo 4Finalmente, subtraímos matriz a matriz. Cada entrada de A\,B^T - 2C é (A\,B^T - 2C)_{ij} = (A\,B^T)_{ij} - (2C)_{ij}. Calculando linha a linha: Linha 1: 1-0=1,\;17-2=15,\;-3-0=-3. Linha 2: -1-14=-15,\;-3-6=-9,\;-5-2=-7. Linha 3: 2-(-8)=10,\;27-10=17,\;-2-1=-3. Logo A\,B^T - 2C = \begin{pmatrix} 1 & 15 & -3 \\[3pt] -15 & -9 & -7 \\[3pt] 10 & 17 & -3 \end{pmatrix}.] Resposta \begin{pmatrix} 1 & 15 & -3 \\[3pt] -15 & -9 & -7 \\[3pt] 10 & 17 & -3 \end{pmatrix}.] MOSTRAR SOLUÇÃO COMPLETAExercícios relacionadosExercício 7 Sejam as matrizes A=\left[\begin{array}{cc}2 & 1 \\\\ 0 & -1 \\\\ 3 & 2\end{array}\right] \quad B=\left[\begin{array}{cc}0 & 1 \\\\ 7 & 3 \\\\ -4 & 5\end{array}\right] \quad C=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ 7 & 3 & 1 \\\\ -4 & 5 & \frac{1}{2}\end{array}\right] \quad D= \left[\begin{array}{ccc}3 & 2 & 1 \\\\ -4 & 6 & 5\end{array}\right] \quad E=\left[\begin{array}{ll}0 & 1 \\\\ 7 & 3\end{array}\right] \quad F=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ -1 & 3 & -2 \\\\ 4 & 4 & -4\end{array}\right] \quad G=\left[\begin{array}{cc}-\frac{3}{7} & \frac{1}{7} \\\\ 1 & 0\end{array}\right] \quad H=\left[\begin{array}{c}4 \\\\ -19 \\\\ 26\end{array}\right] Calcule os seguintes produtos de matrizes D \cdot H Exercício 7 Sejam as matrizes A=\left[\begin{array}{cc}2 & 1 \\\\ 0 & -1 \\\\ 3 & 2\end{array}\right] \quad B=\left[\begin{array}{cc}0 & 1 \\\\ 7 & 3 \\\\ -4 & 5\end{array}\right] \quad C=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ 7 & 3 & 1 \\\\ -4 & 5 & \frac{1}{2}\end{array}\right] \quad D= \left[\begin{array}{ccc}3 & 2 & 1 \\\\ -4 & 6 & 5\end{array}\right] \quad E=\left[\begin{array}{ll}0 & 1 \\\\ 7 & 3\end{array}\right] \quad F=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ -1 & 3 & -2 \\\\ 4 & 4 & -4\end{array}\right] \quad G=\left[\begin{array}{cc}-\frac{3}{7} & \frac{1}{7} \\\\ 1 & 0\end{array}\right] \quad H=\left[\begin{array}{c}4 \\\\ -19 \\\\ 26\end{array}\right] Calcule os seguintes produtos de matrizes G \cdot E Exercício 7 Sejam as matrizes A=\left[\begin{array}{cc}2 & 1 \\\\ 0 & -1 \\\\ 3 & 2\end{array}\right] \quad B=\left[\begin{array}{cc}0 & 1 \\\\ 7 & 3 \\\\ -4 & 5\end{array}\right] \quad C=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ 7 & 3 & 1 \\\\ -4 & 5 & \frac{1}{2}\end{array}\right] \quad D= \left[\begin{array}{ccc}3 & 2 & 1 \\\\ -4 & 6 & 5\end{array}\right] \quad E=\left[\begin{array}{ll}0 & 1 \\\\ 7 & 3\end{array}\right] \quad F=\left[\begin{array}{ccc}0 & 1 & 0 \\\\ -1 & 3 & -2 \\\\ 4 & 4 & -4\end{array}\right] \quad G=\left[\begin{array}{cc}-\frac{3}{7} & \frac{1}{7} \\\\ 1 & 0\end{array}\right] \quad H=\left[\begin{array}{c}4 \\\\ -19 \\\\ 26\end{array}\right] Calcule os seguintes produtos de matrizes A \cdot B^{T}-2 C Exercício 8 Determine a e b para que as matrizes A=\left[\begin{array}{cc}2 & 3 \\\\ -9 & 5\end{array}\right] e B=\left[\begin{array}{cc}a & -1 \\\\ 3 & b\end{array}\right] comutem.