Find the specified change-of-coordinates matrix.
-$\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0$ for all $\mathrm { k }$
A) $1 \mathrm { k } , ( - 1 ) ^ { \mathrm { k } } , ( - 6 ) ^ { \mathrm { k } }$
B) $1 ^ { \mathrm { k } } , ( - 7 ) ^ { \mathrm { k } }$
C) $1 \mathrm { k } , ( - 6 ) ^ { \mathrm { k } }$
D) $( - 1 ) ^ { \mathrm { k } } , 6 ^ { \mathrm { k } }$

Question

Quizplus · Accepted Answer

The characteristic equation of the given recurrence relation is $r^2 + 5r - 6 = 0$, which factors to $(r + 6)(r - 1) = 0$, giving roots $r = -6$ and $r = 1$. The solutions to the recurrence relation are of the form $c_1(-1)^k + c_2(6)^k$, corresponding to the roots of the characteristic equation.

Find the Specified Change-Of-Coordinates Matrix yk+2+5yk+1−6yk=0\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0yk+2​+5yk+1​−6yk​=0

Find the Specified Change-Of-Coordinates Matrix $\mathrm { y } _ { \mathrm { k } + 2 } + 5 \mathrm { y } _ { \mathrm { k } + 1 } - 6 \mathrm { y } _ { \mathrm { k } } = 0$