$$$e^{\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]}$$$
Calculadora relacionada: Calculadora de potência de matriz
Sua entrada
Encontre $$$e^{\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]}$$$.
Solução
Primeiro, diagonalize a matriz (para ver as etapas, consulte calculadora de diagonalização de matrizes).
$$$P = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]$$$
$$$D = \left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]$$$
Encontre o inverso de $$$P$$$: $$$P^{-1} = \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right]$$$ (para ver as etapas, consulte calculadora inversa de matrizes).
Agora, $$$e^{\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]} = e^{\left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]\cdot \left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]\cdot \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right]} = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]\cdot e^{\left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]}\cdot \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right].$$$
A exponencial de uma matriz diagonal é uma matriz cujas entradas diagonais são exponenciadas: $$$e^{\left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]} = \left[\begin{array}{cc}e & 0\\0 & e^{-2}\end{array}\right].$$$
Assim, $$$e^{\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]} = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]\cdot \left[\begin{array}{cc}e & 0\\0 & e^{-2}\end{array}\right]\cdot \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right].$$$
Finalmente, multiplique as matrizes:
$$$\left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]\cdot \left[\begin{array}{cc}e & 0\\0 & e^{-2}\end{array}\right] = \left[\begin{array}{cc}5 e & \frac{2}{e^{2}}\\e & e^{-2}\end{array}\right]$$$ (para ver as etapas, consulte calculadora de multiplicação de matrizes).
$$$\left[\begin{array}{cc}5 e & \frac{2}{e^{2}}\\e & e^{-2}\end{array}\right]\cdot \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right] = \left[\begin{array}{cc}\frac{-2 + 5 e^{3}}{3 e^{2}} & \frac{10 - 10 e^{3}}{3 e^{2}}\\\frac{-1 + e^{3}}{3 e^{2}} & \frac{5 - 2 e^{3}}{3 e^{2}}\end{array}\right]$$$ (para ver as etapas, consulte calculadora de multiplicação de matrizes).
Responder
$$$e^{\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]} = \left[\begin{array}{cc}\frac{-2 + 5 e^{3}}{3 e^{2}} & \frac{10 - 10 e^{3}}{3 e^{2}}\\\frac{-1 + e^{3}}{3 e^{2}} & \frac{5 - 2 e^{3}}{3 e^{2}}\end{array}\right]\approx \left[\begin{array}{cc}4.440246191940667 & -8.609821817408108\\0.860982181740811 & -1.586629080245009\end{array}\right]$$$A