Dot product versus matrix multiplication, is the later a special case of the first?

I seemed to have thoroughly confused myself today...

Long story short, the question is simple. Is matrix multiplication just a special case of the dot product of two sets of vectors when the sets of vectors have the same cardinality and all vectors in both sets have the same length?

I assume the answer is yes from reviewing the computation of matrix multiplication and the dot product.

Solution 1:

Dot product is defined between two vectors.

Matrix product is defined between two matrices.

They are different operations between different objects.

The connection between the two operations that comes to my mind is the following: To calculate the $c_{i,j}$ entry of the matrix $C:=AB$, one takes the dot product of the $i$'th row of the matrix $A$ with the $j$'th column of the matrix $B.$

Solution 2:

What @Pawel said, additionally, though, I would like to add that there is a nice duality between $1\times 2$ matrices and 2d vectors.

3Blue1Brown covers this in the 9th episode of his series 'Essence of Linear Algebra'

Solution 3:

$$ A^TB \ \equiv A \bullet B \iff A \ \text{and} \ B \ \text{are} \ n \times 1 \ \text{matrices}. $$

So you could think of a dot product as a special case of matrix multiplication.