## Square matrices or the form nxn can take an inverse such that the original matrix

## multiplied by its inverse yields the identity matrix with 1's on the diagonal. In

## this R script, we compute the inverse of a matrix "x" but before doing so, check to

## see if the inverse is already in cache memory. Since inverse matrix operations are

## processor intensive, it is beneficial to check in cache for an inverse matrix for a

## matrix "x" that has not changed since the inverse operation was last run. If in cache,

## we can pull the matrix from cache and avoid running the inverse operation over again.

## In this function, we set up the inverse matrix in cache memory.

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

## initialize NULL as the stored value of the inverse matrix

m_inv <- NULL

## if the matrix "x" has changed, we assign the stored value back to NULL

set <- function(y) {

x <<- y

m_inv <<- NULL

}

## get the value of the initial matrix "x"

get <- function() x

## compute the inverse of the matrix and assign it to cache

set_inv <- function(scope) m_inv <<- scope

get_inv <- function() m_inv

## we return values for four variables

list(set = set, get = get,

set_inv = set_inv,

get_inv = get_inv)

}

## We want to compute the inverse of a matrix but before we do that, we check to see if it

## is in cache already

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

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

m_inv <- x$get_inv()

## If the cached value is non-empty, we do not need to run the inverse operation.

## We can simply return the cached matrix.

if(!is.null(m_inv)) {

message("getting cached data")

return(m_inv)

}

## in the event that the cached matrix is null, we need to go ahead and compute the

## matrix inverse. We can then set the value of the inverse matrix to cache.

data <- x$get()

m_inv <- scope(x)

x$set_inv(m_inv)

m_inv

}