What is  matrix A power n if A=(0,-a_a,0)?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to use mathematical induction to calculate `A^n` .

You need to solve the first step, using the following values for n such that:

n = 2 => `A^2 = ((0,-a),(a,0))*((0,-a),(a,0))`  = `((0*0+a*(-a), 0*(-a) + 0*(-a)),(a*0 + 0*a, a*(-a)+0*0))`

`A^2 = ((-a^2,0),(0,-a^2))`

n = 3 => `A^3 = A^2*A = ((-a^2,0),(0,-a^2))*((0,-a),(a,0))` = `((0,-a^3),(-a^3,0))`

n = 4 `=gt A^4 = A^2*A^2 = ((-a^2,0),(0,-a^2))*((-a^2,0),(0,-a^2))` =`((-a^4,0),(0,-a^4))`

Notice that `A^n`  depends on the value of n.

If n is odd, n=2m+1, then `A^n = ((0,-a^(2m+1)),(-a^(2m+1),0))`

`` If n is even, n = 2m, then `A^n = ((-a^(2m),0),(0,-a^(2m))).`

Hence, the matrix `A^n `  may have two forms, depending on n values, such that: `A^n = ((0,-a^(2m+1)),(-a^(2m+1),0)) ;A^n = ((-a^(2m),0),(0,-a^(2m))).`