From 56bb446a57b6a71dd7de24888370d4764e113015 Mon Sep 17 00:00:00 2001
From: Naman Garg <naman@Namans-MacBook-Air.local>
Date: Fri, 24 Oct 2025 05:20:22 +0530
Subject: [PATCH 1/2] defined the functions makeCacheMatrix and cacheSolve

---
 cachematrix.R | 19 ++++++++++++++++++-
 1 file changed, 18 insertions(+), 1 deletion(-)

diff --git a/cachematrix.R b/cachematrix.R
index a50be65aa4..174e109a09 100644
--- a/cachematrix.R
+++ b/cachematrix.R
@@ -4,7 +4,17 @@
 ## Write a short comment describing this function
 
 makeCacheMatrix <- function(x = matrix()) {
-
+  i <- NULL
+  set <- function(y) {
+    x <<- y
+    i <<- NULL
+  }
+  get <- function() x
+  setinverse <- function() {
+    i <<- solve(x)
+  }
+  getinverse <- function() i
+  list(set=set,get=get,setinverse=setinverse,getinverse=getinverse)
 }
 
 
@@ -12,4 +22,11 @@ makeCacheMatrix <- function(x = matrix()) {
 
 cacheSolve <- function(x, ...) {
         ## Return a matrix that is the inverse of 'x'
+  i <- x$getinverse()
+  if (!is.null(i)){
+    message("from the cache")
+    return(i)
+  }
+  i <- x$setinverse()
+  i
 }

From ee4787b53e24cd5e92cede3c697679e9895b1041 Mon Sep 17 00:00:00 2001
From: Naman Garg <naman@Namans-MacBook-Air.local>
Date: Fri, 24 Oct 2025 05:54:10 +0530
Subject: [PATCH 2/2] added comments to the function

---
 cachematrix.R | 9 +++++++--
 1 file changed, 7 insertions(+), 2 deletions(-)

diff --git a/cachematrix.R b/cachematrix.R
index 174e109a09..13c66194f6 100644
--- a/cachematrix.R
+++ b/cachematrix.R
@@ -1,7 +1,10 @@
 ## Put comments here that give an overall description of what your
 ## functions do
 
-## Write a short comment describing this function
+## this function returns a list of functions.
+## the functions within the list all have their scope set within the parent function
+## the functions within the list allow us to cache a matrix, retrieve a cached matrix, 
+## cache inverse of the cached matrix, retrieve cached inverse of the a cached matrix
 
 makeCacheMatrix <- function(x = matrix()) {
   i <- NULL
@@ -18,7 +21,9 @@ makeCacheMatrix <- function(x = matrix()) {
 }
 
 
-## Write a short comment describing this function
+## this function returns the inverse of the cached matrix within the variable
+## the functions first checks if the inverse is cached and sends the cached results back if condition is true
+## if the inverse of the matrix isn't cached the the function solves for the inverse, caches it, and shares back the result
 
 cacheSolve <- function(x, ...) {
         ## Return a matrix that is the inverse of 'x'