A Quick Introduction to Matrix-Matrix Multiplication

Multiplying two matrices together is most easily explained with an example. Once we work through the example, we’ll express what we did in words.

Imagine we have two matrices, A and B. A has 5 rows and 3 columns. B has 3 rows and 4 columns. Multiplying A * B results in C, which has 5 rows and 4 columns.

NewImageThe numbers down the left of each matrix are the row numbers. The ones across the top are the column numbers.

We refer to an element in a matrix by its row and column position: Matrix Name[row, column]. A[1,1] is 1. B[2,3] is 6. C[5,4] is… well, we don’t know what C[5,4] is.

NewImageThe shaded cells are the elements referred to.

To calculate C[1,1], we need the first row of A and the first column of B.

NewImage

We need to take the dot product of the first row of A and the first column of B. The dot product is represented by a dot: x • y. Dot product means multiplying each element of A’s first row by the corresponding element of B’s first column, then adding up the result of the multiplications. Put another way:

A's Row 1 • B's Column 1 = A[1,1] * B[1,1] + A[1,2] * B[2,1] + A[1,3] * B[3,1] 

It’s easier to calculate if we write it down. First, we’ll take just the row and column we need.

NewImageThe numbers down the left are the row numbers.
The ones across the top are the column numbers.

Next, we’ll write B’s first column underneath A’s first row, lining up A’s columns with B’s rows.

NewImageThe numbers across the top represent column numbers for A and row numbers for B.

Then we’ll multiply the numbers in each column and write each result underneath. 1 * 12 is 12, 2 * 8 is 16, and 3 * 4 is 12.NewImageFinally, we add them together. 12 + 16 + 12 = 40NewImage

And put the result in C[1,1].

NewImage

The remaining elements of C are calculated in a similar manner. C[1,2] is the dot product of the first row of A and the second column of B. C[1, 3] is the dot product of the first row of A and the third column of B.

Even though A only has three columns, we’re taking the dot product of rows of A and columns of B. Since B has another column, we take the dot product of the first row of A and the fourth column of B to calculate C[1, 4]. If B had additional columns, we would continue calculating dot products using the first row of A until we ran out of columns.

The remaining rows of A follow the same pattern. C[2,1] is the dot product of the second row of A and the first column of B. C[3,2] is the dot product of the third row of A and the second row of B. And so forth.

The general rule is C[i, j] equals the dot product of the i-th row of A and the j-th column of B. We’ll do one more example. C[5, 3] is the dot product of the fifth row of A and the third column of B.

NewImage

The dot product of A’s row 5 and B’s row 3 is 244.

NewImage

So C[5, 3] is 244. Let’s fill in the rest of C.

NewImage

Matrix Multiplication Formula

Taking what we learned from the above example:

If A is a matrix with M rows and P columns and B is a matrix with P rows and N columns, A * B results in a matrix C. C has M rows and N columns. Each element in C, C[i, j], is the dot product of the i-th row of A and the j-th column of B. Two matrices can be multiplied if and only if A has the same number of columns that B has rows.

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