Algorithm to multiply two matrices

1. Well, first you should create your own class for matrices. Your current ad-hoc representation is inefficient and lacks abstraction.
   Options include:

   • Row-major or column-major.
   • Restricted to specific dimensions / square matrices.
   • Using an array of pointers to rows/colums or explicit mapping of coordinates to ids.

2. Is there a reason you insist that the result-matrix should be square? Seems gratuitious.

3. There's no advantage to modifying the first input instead of returning a new matrix. So, make both arguments constant and return the result instead.

4. If you don't need an object any longer after copying it, might moving be more appropriate, as it's potentially more efficient and less likely to throw?

Matrix multiplication, while it seems trivial to implement from the definition, the naive implementation you are using is actually slow for anything but small matrices.

You really should look into more efficient algorithms for matrix multiplication, a good place to start is the Wikipedia page here: Matrix multiplication algorithm.

A matrix multiplication algorithm usually involves three for loops. The fact that you are using two for loops and use some obtuse logic to reset indices point to unclear thinking on your part. As a matter of fact, there is an error in your logic and is easily exposed if you allow for multiplication of matrices that don't result in square matrices.

If you change

if (matrix1.size() == matrix2.at(0).size() && matrix1.at(0).size() == matrix2.size())

to

if (matrix1.at(0).size() == matrix2.size())

you allow multiplication of matrices that don't result in a square matrix. However, if you multiply the following matrices with just that change,

std::vector<std::vector<int>> matrix1{ { 1,0,2,1 },{ -1,3,1,2 } };
std::vector<std::vector<int>> matrix2{ { 3,1 },{ 2,1 },{ 1,0 } };
function(matrix2, matrix1);

you will notice that the resultant is not correct. It ends up being

2 3 7 0 
1 3 5 0 
1 0 2 0 

Please note that the correct one needs to be:

2 3 7 5 
1 3 5 4 
1 0 2 1 

By changing the function to

void function(std::vector<std::vector<int>>& matrix1,
              const std::vector<std::vector<int>>& matrix2)
{
    std::vector<std::vector<int>> matrix(matrix1.size(), std::vector<int>(matrix2.at(0).size()));
    if (matrix1.at(0).size() == matrix2.size())
    {
        for (size_t i = 0; i < matrix1.size(); ++i)
        {
            for (size_t j = 0; j < matrix1.at(0).size(); ++j)
            {
               for (size_t k = 0; k < matrix2.at(0).size(); ++k )
               {
                  matrix.at(i).at(k) += matrix1.at(i).at(j) * matrix2.at(j).at(k);
               }
            }
        }
    }
    else
        throw std::invalid_argument("Invalid syntax");
    matrix1 = std::move(matrix);
}

that error is fixed.

Please note that I changed the type of i and j to size_t from int.