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

# makes a cache matrix from a given matrix

# assign the value NULL for the first initialization

cacheMatrix <- NULL

# define the function named 'setMatrix'

setMatrix <- function(y) {

x <<- y

cacheMatrix <<- NULL

}

# define the method named 'getMatrix'

getMatrix <- function() x

# define the method named 'setCache'

setCache <- function(inverse) cacheMatrix <<- inverse

# define the method named 'getCache'

# return the cached inverse of 'x'

getCache <- function() cacheMatrix

list(setMatrix = setMatrix,

getMatrix = getMatrix,

setCache = setCache,

getCache = getCache)

}

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

# return the inverse of a given matrix utilizing the cache

cacheMatrix <- x$getCache()

# if the content is not null then: return the result

if (!is.null(cacheMatrix)) {

message("loading cache matrix...")

return(cacheMatrix)

}

# if the content is empty:

else {

dMatrix <- x$getMatrix()

cacheMatrix <- solve(dMatrix, ...)

x$setCache(cacheMatrix)

return(cacheMatrix)

}

}