matrix multiplication algorithm time complexity

Question

I came up with this algorithm for matrix multiplication. I read somewhere that matrix multiplication has a time complexity of o(n^2). But I think my this algorithm will give o(n^3). I don't know how to calculate time complexity of nested loops. So please correct me.

for i=1 to n
   for j=1 to n    
     c[i][j]=0
     for k=1 to n
         c[i][j] = c[i][j]+a[i][k]*b[k][j]

Answer 1

Using linear algebra, there exist algorithms that achieve better complexity than the naive O(n³). Solvay Strassen algorithm achieves a complexity of O(n^2.807) by reducing the number of multiplications required for each 2x2 sub-matrix from 8 to 7.

The fastest known matrix multiplication algorithm is Coppersmith-Winograd algorithm with a complexity of O(n^2.3737). Unless the matrix is huge, these algorithms do not result in a vast difference in computation time. In practice, it is easier and faster to use parallel algorithms for matrix multiplication.