Diagonaliza las siguientes matrices si es posible:

(A) $A=\left(\begin{array}{cc}3& 4\\ 5& 2\end{array}\right)$

(B) $B=\left(\begin{array}{ccc}-2& 2& 0\\ -6& 5& 0\\ 0& 0& 0\end{array}\right)$

(C) $C=\left(\begin{array}{ccc}1& 0& 0\\ 1& -1& 1\\ 0& 0& 1\end{array}\right)$

(K) $K=\left(\begin{array}{ccc}3& -3& 4\\ -3& 3& 4\\ -2& -2& 0\end{array}\right)$

(E) $E=\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& -1& -1\\ 0& 0& -1& -1\end{array}\right)$

(F) $F=\left(\begin{array}{cc}0& 2\\ 0& 0\end{array}\right)$

(G) $G=\left(\begin{array}{ccc}2& -1& -1\\ 4& 3& -4\\ -3& -1& 4\end{array}\right)$

(H) $H=\left(\begin{array}{ccc}-2& -2& -1\\ -2& 1& 2\\ 1& -2& -4\end{array}\right)$

(I) $I=\left(\begin{array}{ccc}-1& 0& -1\\ 0& 0& 0\\ 1& 0& 1\end{array}\right)$

(J) $J=\left(\begin{array}{cccc}5& 0& 0& 0\\ 0& 3& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 5\end{array}\right)$

Para las matrices del ejercicio anterior, calcula las siguientes potencias:

$A^5,B^9,C^{21},K^2,E^{15},F^{88},G^5,H^4,I^{47},J^6$

(A) $D=\left(\begin{array}{cc}7& 0\\ 0& -2\end{array}\right),P=\left(\begin{array}{cc}1& -4\\ 1& 5\end{array}\right)$

(B) $D=\left(\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right),P=\left(\begin{array}{ccc}1& 2& 0\\ 2& 3& 0\\ 0& 0& 1\end{array}\right)$

(C) $D=\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),P=\left(\begin{array}{ccc}0& -1& 2\\ 1& 0& 1\\ 0& 1& 0\end{array}\right)$

(K) No es diagonalizable (faltan autovalores).

(E) $D=\left(\begin{array}{cccc}-2& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),P=\left(\begin{array}{cccc}0& 1& 0& -1\\ 0& 1& 0& 1\\ 1& 0& -1& 0\\ 1& 0& 1& 0\end{array}\right)$

(F) No es diagonalizable (faltan autovectores).

(G) $D=\left(\begin{array}{ccc}5& 0& 0\\ 0& 3& 0\\ 0& 0& 1\end{array}\right),P=\left(\begin{array}{ccc}1& 1& 1\\ -8& -2& 0\\ 5& 1& 1\end{array}\right)$

(H) $D=\left(\begin{array}{ccc}-3& 0& 0\\ 0& -3& 0\\ 0& 0& 1\end{array}\right),P=\left(\begin{array}{ccc}1& 2& 1\\ 0& 1& -2\\ 1& 0& 1\end{array}\right)$

(I) No es diagonalizable (faltan autovectores).

(J) $D=\left(\begin{array}{cccc}5& 0& 0& 0\\ 0& 3& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 5\end{array}\right),P=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$

• $A^5=\left(\begin{array}{cc}9323& 7484\\ 9355& 7452\end{array}\right)$

• $B^9=\left(\begin{array}{ccc}-1532& 1022& 0\\ -3066& 2045& 0\\ 0& 0& 0\end{array}\right)$

• $C^{21}=\left(\begin{array}{ccc}1& 0& 0\\ 1& -1& 1\\ 0& 0& 1\end{array}\right)$

• $K^2=\left(\begin{array}{ccc}10& -26& 0\\ -26& 10& 0\\ 0& 0& -16\end{array}\right)$

• $E^{15}=\left(\begin{array}{cccc}16384& 16384& 0& 0\\ 16384& 16384& 0& 0\\ 0& 0& -16384& -16384\\ 0& 0& -16384& -16384\end{array}\right)$

• $F^{88}=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$

• $G^5=\left(\begin{array}{ccc}-538& -121& 539\\ 5764& 243& -5764\\ -3663& -121& 3664\end{array}\right)$

• $H^4=\left(\begin{array}{ccc}61& 40& 40\\ 20& 1& -40\\ -20& 40& 101\end{array}\right)$

• $I^{47}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

• $J^6=\left(\begin{array}{cccc}15625& 0& 0& 0\\ 0& 729& 0& 0\\ 0& 0& 64& 0\\ 0& 0& 0& 15625\end{array}\right)$