Question Number 100994 by mathmax by abdo last updated on 29/Jun/20

let A = (((2 1)),((1 3)) ) 1) calculate A^n 2) find e^A ,e^(−A) 3)determine ch(A) and sh(A) is ch^2 A−sh^2 A =1?

Answered by mathmax by abdo last updated on 01/Jul/20

1) the caracteristic polynom of A is p_c (x)=det(A−xI) = determinant (((2−x 1)),((1 3−x))) =(x−2)(x−3)−1 =x^2 −5x+6−1 =x^2 −5x +5 P_c (x)=0 ⇔x^2 −5x +5 =0 →Δ =25−20 =5 ⇒α=((5+(√5))/2) and β=((5−(√5))/2) we have x^(n ) =qP_c (x) +u_n x +v_n ⇒ { ((α^n =u_n α +v_n )),((β^n =u_n β +v_n )) :} ⇒u_n =((α^n −β^n )/(α−β)) =(1/(√5)){ (((5+(√5))/2))^n −(((5−(√5))/2))^n } v_n =α^n −αu_n =α^n −α×((α^n −β^n )/(α−β)) =((α^(n+1) −β α^n −α^(n+1) +αβ^n )/(α−β)) =((αβ^n −βα^n )/(α−β)) =((((5+(√5))/2)(((5−(√5))/2))^n −((5−(√5))/2)(((5+(√5))/2))^n )/(√5)) we have A^n =u_n A +v_n I (by cayley hamilton) ⇒ A^n =(1/(√5)){(((5+(√5))/2))^n −(((5−(√5))/2))^n } (((2 1)),((1 3)) ) +((((5+(√5))/2)(((5−(√5))/2))^n −((5−(√5))/2)(((5+(√5))/2))^n )/(√5)) (((1 0)),((0 1)) )