## These functions implement an object in R representing a matrix with the

## ability to cache its own inverse when using the cacheSolve function.

## makeCacheMatrix makes an R object to cache the inverse for a given matrix

makeCacheMatrix <- function(x = matrix()) {

# initially we do not know the inverse

i <- NULL

# define the functions this object implements

set <- function(y) {

x <<- y

# must reset cached inverse when x changes

i <<- NULL

}

get <- function() x

setinverse <- function(inverse) i <<- inverse

getinverse <- function() i

# return the object as a list

list(set = set, get = get,

setinverse = setinverse,

getinverse = getinverse)

}

## cacheSolve solves the inverse for the matrix object, unless it is cached

cacheSolve <- function(x, ...) {

# if it is cached, then just return it

i <- x$getinverse()

if(!is.null(i)) {

# for debugging: message("getting cached data")

return(i)

}

# otherwise, solve and cache it for future use

data <- x$get()

i <- solve(data, ...)

x$setinverse(i)

## Return a matrix that is the inverse of 'x'

i

}