Update problems-matrix.md

docs/Matrix/problems-matrix.md

@@ -1,130 +1,134 @@
 ---
 id: matrix-problems
-title:  Matrix Practice Problems
+title: Matrix Practice Problems
 sidebar_label: Matrix Practice Problems
 sidebar_position: 2
-description: "A matrix is a two-dimensional data structure consisting of rows and columns, where each element is identified by its row and column index. It is commonly used in various fields, including mathematics, computer science, and data analysis, to represent and manipulate structured data. "
-tags: [dsa, data-structures, Matrix ]
+description: "A matrix is a two-dimensional data structure consisting of rows and columns, where each element is identified by its row and column index. It is commonly used in various fields, including mathematics, computer science, and data analysis, to represent and manipulate structured data."
+tags: [dsa, data-structures, Matrix]
 ---
 
-## Sort the given matrix
+## Sort the Given Matrix
 
-Given a n x n matrix. The problem is to sort the given matrix in strict order. Here strict order means that the matrix is sorted in a way such that all elements in a row are sorted in increasing order and for row ‘i’, where ```1 <= i <= n-1```, the first element of row ‘i’ is greater than or equal to the last element of row ‘i-1’.
+Given an `n x n` matrix, the problem is to sort the given matrix in strict order. **Strict order** means that the matrix is sorted such that all elements in each row are sorted in increasing order. For row `i`, where `1 <= i <= n-1`, the first element of row `i` is greater than or equal to the last element of row `i-1`.
 
 **Examples:** 
 ```
-Input : mat[][] = { {5, 4, 7},
-                    {1, 3, 8},
-                    {2, 9, 6} }
-Output : 1 2 3
-         4 5 6
-         7 8 9
+Input: mat[][] = {{5, 4, 7},
+                  {1, 3, 8},
+                  {2, 9, 6}}
+
+Output: 1 2 3
+        4 5 6
+        7 8 9
 ```
-## Solution
+
+### Solution
 ```python
-# Python program for the above approach
-# driver code
-v = [[5,4,7], [1,3,8], [2,9,6]]
+# Python program for sorting the matrix in strict order
+
+# Input matrix
+v = [[5, 4, 7], [1, 3, 8], [2, 9, 6]]
 n = len(v)
-
-x = []
-for i in range(n):
-    for j in range(n):
-        x.append(v[i][j])
-
+
+# Flatten the matrix into a list
+x = [v[i][j] for i in range(n) for j in range(n)]
+
+# Sort the flattened list
 x.sort()
+
+# Insert sorted elements back into the matrix
 k = 0
 for i in range(n):
     for j in range(n):
         v[i][j] = x[k]
         k += 1
-
+
+# Print the sorted matrix
 print("Sorted Matrix will be: ")
 for i in range(n):
     for j in range(n):
         print(v[i][j], end=" ")
     print("")
-
-# THIS CODE IS CONTRIBUTED BY Dhairya Gothi(dhairyagothi)
 ```
 
-## Output
-Sorted Matrix Will be:
+### Output:
 ```
+Sorted Matrix will be: 
 1 2 3 
 4 5 6 
-7 8 9 
+7 8 9
 ```
-**Time Complexity:** O(n2log2n), O(n*n) for traversing, and O(n2log2n) for sorting the vector x, which has a size of n2. So overall time complexity is O(n2log2n).
-**Auxiliary Space:** O(n*n), For vector.
 
-## Program for scalar multiplication of a matrix
+**Time Complexity:**  
+- Sorting: $O(n^2 \log n)$ where $n$ is the number of elements in the matrix.  
+- Traversing the matrix to collect elements: $O(n^2)$.  
+So, the overall time complexity is $O(n^2 \log n)$.
+
+**Auxiliary Space:**  
+- $O(n^2)$ for storing the matrix elements in a flattened list.
 
-Given a matrix and a scalar element k, our task is to find out the scalar product of that matrix. 
+---
+
+## Program for Scalar Multiplication of a Matrix
+
+Given a matrix and a scalar element $k$, our task is to compute the scalar product of the matrix, where each element is multiplied by $k$.
 
 **Examples:** 
 ```
-Input : mat[][] = {{2, 3}
-                   {5, 4}}
-        k = 5
-Output : 10 15 
-         25 20 
-We multiply 5 with every element.
-
-Input : 1 2 3 
-        4 5 6
-        7 8 9
-        k = 4
-Output :  4 8  12
-          16 20 24
-          28 32 36 
-The scalar multiplication of a number k(scalar), multiply it on every entry in the matrix. and a matrix A is the matrix kA.
- ```
+Input: mat[][] = {{2, 3},
+                  {5, 4}}
+       k = 5
+
+Output: 10 15
+        25 20
+
+Input: mat[][] = {{1, 2, 3},
+                  {4, 5, 6},
+                  {7, 8, 9}}
+       k = 4
+
+Output: 4  8  12
+        16 20 24
+        28 32 36
+```
 
+### Solution
 ```python
-# Python 3 program to find the scalar 
-# product of a matrix
-
-# Size of given matrix
-N = 3
-
-def scalarProductMat( mat, k):
-
-    # scalar element is multiplied 
-    # by the matrix
-    for i in range( N):
-        for j in range( N): 
-            mat[i][j] = mat[i][j] * k     
-
+# Python program for scalar multiplication of a matrix
+
+def scalarProductMat(mat, k):
+    N = len(mat)  # Get the size of the matrix
+    # Multiply each element of the matrix by k
+    for i in range(N):
+        for j in range(N):
+            mat[i][j] *= k
+
 # Driver code
 if __name__ == "__main__":
-
-    mat = [[ 1, 2, 3 ],
-           [ 4, 5, 6 ],
-           [ 7, 8, 9 ]]
+    mat = [[1, 2, 3],
+           [4, 5, 6],
+           [7, 8, 9]]
     k = 4
-
+
+    # Perform scalar multiplication
     scalarProductMat(mat, k)
-
-    # to display the resultant matrix
-    print("Scalar Product Matrix is : ")
-    for i in range(N):
-        for j in range(N):
-            print(mat[i][j], end = " ")
-        print()
-
-# This code is contributed by dhairya
+
+    # Print the result
+    print("Scalar Product Matrix is: ")
+    for row in mat:
+        print(" ".join(map(str, row)))
 ```
-## Output: 
+
+### Output:
 ```
-Scalar Product Matrix is : 
-4 8 12 
-16 20 24 
+Scalar Product Matrix is:
+4 8 12
+16 20 24
 28 32 36
 ```
-
-
-**ime Complexity:** O(n2),
 
-**Auxiliary Space:** O(1), since no extra space has been taken.
+**Time Complexity:**  
+- $O(n^2)$, where $ n $ is the dimension of the matrix.
 
+**Auxiliary Space:**  
+- $O(1)$, since the operation is performed in place.