Unfortunately, x^(-1) is not the inverse of x:
x<-matrix(c(2,4,4,5),nrow=2)
x
# [,1] [,2]
# [1,] 2 4
# [2,] 4 5
x^(-1)
# [,1] [,2]
# [1,] 0.50 0.25
# [2,] 0.25 0.20
i.e. it is the matrix which you get by applying the operation
(...)^(-1) to each element of x.
In R, the inverse of a non-singular matrix is (somewhat obscurely)
denoted by solve(x):
solve(x)
# [,1] [,2]
# [1,] -0.8333333 0.6666667
# [2,] 0.6666667 -0.3333333
solve(x)%*%x
# [,1] [,2]
# [1,] 1 1.110223e-16
# [2,] 0 1.000000e+00
(Note the slight rounding error); whereas
(x^(-1))%*%x
# [,1] [,2]
# [1,] 2.0 3.25
# [2,] 1.3 2.00