update doc/cheatsheet/matrix.md

Author: yennanliu

doc/cheatsheet/matrix.md

@@ -14,6 +14,8 @@
     - Spiral Matrix I
         - LC 54
         - LC 59
+    - Sparse matrix product
+        - LC 311
 
 - Algorithm
     - [fucking algorithm : 二维数组的花式遍历技巧](https://labuladong.github.io/algo/2/20/26/)
@@ -401,4 +403,67 @@ class Solution(object):
         res = []
         dfs(matrix, target, 0, 0)
         return True in res
+```
+
+### 2-7) Sparse Matrix Multiplication
+
+```java
+// java
+// LC 311
+// V0
+// IDEA : ARRAY OP (fix by gpt)
+/**
+*  Why there is 3 loop ?
+*
+*   -> Matrix Multiplication: Ci,j = Sigma(Aik * Bkj)
+*
+*   -> so we have 3 layer loop as below:
+*    - i : Iterates over the rows of  A  (outer loop).
+*    - j : Iterates over the columns of  B  (second loop).
+*    - k : Iterates over the “shared dimension” (columns of  A  or rows of  B ) to compute the dot product (inner loop).
+*
+*
+*  ->
+*
+*  The Role of the Loops
+*    1.  Outer loop ( i ): Iterates over the rows of mat1 to calculate each row of the result matrix.
+*    2.  Middle loop ( j ): Iterates over the columns of mat2 to compute each element in a row of the result matrix.
+*    3.  Inner loop ( k ): Iterates over the “shared dimension” to compute the dot product of the  i^{th}  row of mat1 and the  j^{th}  column of mat2.
+*
+*
+* ->  Why the Inner Loop ( k ) Exists ?
+*
+*    -> The inner loop is necessary
+*       because each element of the result matrix
+*       is computed as the dot product of a
+*       row from mat1 and a column from mat2.
+*       Without this loop, the computation of the dot product would be incomplete.
+*/
+public static int[][] multiply(int[][] mat1, int[][] mat2) {
+    // Edge case: Single element matrices
+    if (mat1.length == 1 && mat1[0].length == 1 && mat2.length == 1 && mat2[0].length == 1) {
+        return new int[][]{{mat1[0][0] * mat2[0][0]}};
+    }
+
+    int l_1 = mat1.length;    // Number of rows in mat1
+    int w_1 = mat1[0].length; // Number of columns in mat1 (and rows in mat2)
+
+    int w_2 = mat2[0].length; // Number of columns in mat2
+
+    // Initialize the result matrix
+    int[][] res = new int[l_1][w_2];
+
+    // Perform matrix multiplication
+    for (int i = 0; i < l_1; i++) {
+        for (int j = 0; j < w_2; j++) {
+            int sum = 0; // Sum for res[i][j]
+            for (int k = 0; k < w_1; k++) {
+                sum += mat1[i][k] * mat2[k][j];
+            }
+            res[i][j] = sum;
+        }
+    }
+
+    return res;
+}
 ```