From 5db54a7a0aad828af68f9b98a4cff9ec53d48498 Mon Sep 17 00:00:00 2001
From: Michael Nowell <michaelnowell84@yahoo.com>
Date: Mon, 6 Nov 2023 19:11:50 +0800
Subject: [PATCH 1/2] First Save

---
 cachematrix.R | 1 +
 1 file changed, 1 insertion(+)

diff --git a/cachematrix.R b/cachematrix.R
index a50be65aa44..088ef03aa1d 100644
--- a/cachematrix.R
+++ b/cachematrix.R
@@ -1,3 +1,4 @@
+##Second GIT Project
 ## Put comments here that give an overall description of what your
 ## functions do
 

From b2f7a23315da1b69f0a057d6256e56637d48080f Mon Sep 17 00:00:00 2001
From: Michael Nowell <michaelnowell84@yahoo.com>
Date: Mon, 6 Nov 2023 19:28:16 +0800
Subject: [PATCH 2/2] Matrix and invertible function

---
 cachematrix.R | 67 +++++++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 57 insertions(+), 10 deletions(-)

diff --git a/cachematrix.R b/cachematrix.R
index 088ef03aa1d..ba79a7e2712 100644
--- a/cachematrix.R
+++ b/cachematrix.R
@@ -1,16 +1,63 @@
-##Second GIT Project
-## Put comments here that give an overall description of what your
-## functions do
-
-## Write a short comment describing this function
-
 makeCacheMatrix <- function(x = matrix()) {
-
+  ## @x: a square invertible matrix
+  ## return: a list containing functions to
+  ##              1. set the matrix
+  ##              2. get the matrix
+  ##              3. set the inverse
+  ##              4. get the inverse
+  ##         this list is used as the input to cacheSolve()
+  
+  inv = NULL
+  set = function(y) {
+    # use `<<-` to assign a value to an object in an environment 
+    # different from the current environment. 
+    x <<- y
+    inv <<- NULL
+  }
+  get = function() x
+  setinv = function(inverse) inv <<- inverse 
+  getinv = function() inv
+  list(set=set, get=get, setinv=setinv, getinv=getinv)
 }
 
+cacheSolve <- function(x, ...) {
+  ## @x: output of makeCacheMatrix()
+  ## return: inverse of the original matrix input to makeCacheMatrix()
+  
+  inv = x$getinv()
+  
+  # if the inverse has already been calculated
+  if (!is.null(inv)){
+    # get it from the cache and skips the computation. 
+    message("getting cached data")
+    return(inv)
+  }
+  
+  # otherwise, calculates the inverse 
+  mat.data = x$get()
+  inv = solve(mat.data, ...)
+  
+  # sets the value of the inverse in the cache via the setinv function.
+  x$setinv(inv)
+  
+  return(inv)
+}
 
-## Write a short comment describing this function
+##Test function
 
-cacheSolve <- function(x, ...) {
-        ## Return a matrix that is the inverse of 'x'
+test = function(mat){
+  ## @mat: an invertible matrix
+  
+  temp = makeCacheMatrix(mat)
+  
+  start.time = Sys.time()
+  cacheSolve(temp)
+  dur = Sys.time() - start.time
+  print(dur)
+  
+  start.time = Sys.time()
+  cacheSolve(temp)
+  dur = Sys.time() - start.time
+  print(dur)
 }
+  
\ No newline at end of file