Introduction  This second programming assignment will require you to write an R function is able to cache potentially time-consuming computations. For example, taking the mean of a numeric vector is typically a fast operation. However, for a very long vector, it may take too long to compute the mean, especially if it has to be computed repeatedly (e.g. in a loop). If the contents of a vector are not changing, it may make sense to cache the value of the mean so that when we need it again, it can be looked up in the cache rather than recomputed. In this Programming Assignment will take advantage of the scoping rules of the R language and how they can be manipulated to preserve state inside of an R object.   Example: Caching the Mean of a Vector  In this example we introduce the  <<-  operator which can be used to assign a value to an object in an environment that is different from the current environment. Below are two functions that are used to create a special object that stores a numeric vector and cache's its mean.  The first function,  makeVector  creates a special "vector", which is really a list containing a function to  1. set the value of the vector  2. get the value of the vector  3. set the value of the mean  4. get the value of the mean  makeVector <- function(x = numeric()) {         m <- NULL         set <- function(y) {                 x <<- y                 m <<- NULL         }         get <- function() x         setmean <- function(mean) m <<- mean         getmean <- function() m         list(set = set, get = get,              setmean = setmean,              getmean = getmean) }   The following function calculates the mean of the special "vector" created with the above function. However, it first checks to see if the mean has already been calculated. If so, it  get s the mean from the cache and skips the computation. Otherwise, it calculates the mean of the data and sets the value of the mean in the cache via the  setmean  function. cachemean <- function(x, ...) {         m <- x$getmean()         if(!is.null(m)) {                 message("getting cached data")                 return(m)         }         data <- x$get()         m <- mean(data, ...)         x$setmean(m)         m }   Assignment: Caching the Inverse of a Matrix  Matrix inversion is usually a costly computation and their may be some benefit to caching the inverse of a matrix rather than compute it repeatedly (there are also alternatives to matrix inversion that we will not discuss here). Your assignment is to write a pair of functions that cache the inverse of a matrix.  Write the following functions:  1.  makeCacheMatrix : This function creates a special "matrix" object that can cache its inverse.  2.  cacheSolve : This function computes the inverse of the special "matrix" returned by  makeCacheMatrix  above. If the inverse has already been calculated (and the matrix has not changed), then the  cachesolve  should retrieve the inverse from the cache.   Computing the inverse of a square matrix can be done with the  solve  function in R. For example, if  X  is a square invertible matrix, then  solve(X)  returns its inverse.  For this assignment, assume that the matrix supplied is always invertible.  In order to complete this assignment, you must do the following:  1. Clone the GitHub repository containing the stub R files at https://github.com/rdpeng/ProgrammingAssignment2  2. Edit the R file contained in the git repository and place your solution in that file (please do not rename the file).  3. Commit your completed R file into YOUR git repository and push your git branch to your GitHub account.  4. Submit to Coursera the URL to your GitHub repository that contains the completed R code for the assignment.   Grading  This assignment will be graded via peer assessment.