Question

Seja A a matriz $\left[\begin{array}{ccc}0& \frac{1}{2}\\2&0\end{array}\right]$, sabe-se que $A^{n}$ = A · A · A· ... · A (nº de vezes). Então, determine o determinante da matriz S = A + $A^{2}$ + $A^{3}$ + ... + $A^{11}$

Answer 1

$A= \left[\begin{array}{ccc}0& \frac{1}{2}\\2&0\end{array}\right]\\ \\ A^2=\left[\begin{array}{ccc}0& \frac{1}{2}\\2&0\end{array}\right]\left[\begin{array}{ccc}0& \frac{1}{2}\\2&0\end{array}\right]=\left[\begin{array}{ccc}1& 0\\0&1\end{array}\right]$

Entonces

$S=A+\underbrace{A^2}_{I}+\underbrace{A^3}_{A}+\underbrace{A^4}_{I}+\cdots+\underbrace{A^{11}}_{A}\\ \\ S=A+I+A+I+\cdots A+I+A\\ \\ S=6A+5I\\ \\ S=\left[\begin{matrix} 0&3\\12&0 \end{matrix}\right]+\left[\begin{matrix} 5&0\\0&5 \end{matrix}\right]\\ \\ \\ S=\left[\begin{matrix} 5&3\\12&5 \end{matrix}\right]$