# [Matrix] x [Matrix]

Although matrices are not first class citizens in Power Query, quite often a business problem comes along which might greatly be simplified by thinking of it in matrix terms.

You can of course imitate matrices in Power BI by means of tables, two dimensional lists or records but this approach doesn’t come with any matrix related functions. So as always you have to roll up your sleeves and roll out your own set of needed operations.

I would like to share with you an approach of doing matrix multiplication which can be translated into other algebra operations.

We’ll start by reviewing how matrix multiplication works from the diagram below.

If we break down the problem, we can think of two matrices A and B, in M(Power Query) terms as two tables with matching size of rows in matrix A and columns in matrix B respectively. Then the matrix multiplication operation can be reduced to a problem of iterating through all row and column combinations and computing the dot product of equally sized vectors.

The dot product in M can be performed by using List.Accumulate which aggregates the product of a List.Zip which in turn pulls the respective elements of two vectors into a tuples. Here is the code.

```dotProduct = (v1,v2) =>
List.Accumulate(
List.Zip({v1,v2}),0,(agg,_)=> agg+(_{1}*_{0}))
```

Next, we breakdown the matrix A into rows – Table.ToRows(A) – and matrix B into columns Table.ToColumns(B). And iterate through all the combinations of row vectors and column vectors in order to accumulate the results into a nested list of dot products.

Check the code below for use of nested List.Accumulate.

```
List.Accumulate(Table.ToRows(A),
{},
(_,ai)=> _&{ List.Accumulate(Table.ToColumns(B),
{},(_,bj)=> _&{ dotProduct(ai,bj) })
})

```

Putting the two operations together results in the following final function: