(Simon & Blume - Exercício 8.5) Às vezes acontece de A B=B A. Cheque isso para A=\left[\begin{array}{ll}2 & 1 \\\\ 1 & 2\end{array}\right] e B=\left[\begin{array}{cc}3 & -4 \\\\ -4 & 3\end{array}\right]. Passo 1O enunciado pede para verificar se as matrizes A=\begin{pmatrix}2 & 1\\[4pt]1 & 2\end{pmatrix} \quad\text{e}\quad B=\begin{pmatrix}3 & -4\\[4pt]-4 & 3\end{pmatrix} comutam, isto é, se AB=BA. Vamos usar a definição de multiplicação de matrizes 2\times2.] Passo 2Calcular AB. Pelas regras de multiplicação, AB =\begin{pmatrix}2 & 1\\\ 1 & 2\end{pmatrix} \begin{pmatrix}3 & -4\\\ -4 & 3\end{pmatrix} =\begin{pmatrix} 2\cdot3 + 1\cdot(-4) & 2\cdot(-4) + 1\cdot3\\[6pt] 1\cdot3 + 2\cdot(-4) & 1\cdot(-4) + 2\cdot3 \end{pmatrix} =\begin{pmatrix}2 & -5\\\ -5 & 2\end{pmatrix}.] Passo 3Calcular BA. Similarmente, BA =\begin{pmatrix}3 & -4\\\ -4 & 3\end{pmatrix} \begin{pmatrix}2 & 1\\\ 1 & 2\end{pmatrix} =\begin{pmatrix} 3\cdot2 + (-4)\cdot1 & 3\cdot1 + (-4)\cdot2\\[6pt] (-4)\cdot2 + 3\cdot1 & (-4)\cdot1 + 3\cdot2 \end{pmatrix} =\begin{pmatrix}2 & -5\\\ -5 & 2\end{pmatrix}.] Passo 4Comparando, vemos que \,AB=\begin{pmatrix}2 & -5\\\ -5 & 2\end{pmatrix} = BA. RespostaAs matrizes A e B comutam, ou seja, AB=BA. MOSTRAR SOLUÇÃO COMPLETAExercícios relacionadosSejam A=\left[\begin{array}{ccc}2 & 3 & 1 \\\\ 0 & -1 & 2\end{array}\right], B=\left[\begin{array}{ccc}0 & 1 & -1 \\\\ 4 & -1 & 2\end{array}\right], C=\left[\begin{array}{cc}1 & 2 \\\\ 3 & -1\end{array}\right], D=\left[\begin{array}{ll}2 & 1 \\\\ 1 & 1\end{array}\right], E=\left[\begin{array}{c}1 \\\\ -1\end{array}\right] Verifique que C D \neq D C. Sejam A=\left[\begin{array}{ccc}2 & 3 & 1 \\\\ 0 & -1 & 2\end{array}\right], B=\left[\begin{array}{ccc}0 & 1 & -1 \\\\ 4 & -1 & 2\end{array}\right], C=\left[\begin{array}{cc}1 & 2 \\\\ 3 & -1\end{array}\right], D=\left[\begin{array}{ll}2 & 1 \\\\ 1 & 1\end{array}\right], E=\left[\begin{array}{c}1 \\\\ -1\end{array}\right] Verifique que (D A)^{T}=A^{T} D^{T}. (Simon & Blume - Exercício 8.7) Mostre que A=\left[\begin{array}{ll}-1 & 2 \\\\ -1 & 2\end{array}\right] e B=\left[\begin{array}{cc}3 & 6 \\\\ -1 & -2\end{array}\right] são idempotentes. (Simon & Blume - Exercício 8.9) Quantas matrizes de permutação n \times n existem?