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May 23rd

May 23rd

# Matrix Operations

Matrix - matrix multiplication

The first thing to note is that matrix-matrix multiplication is NOT d one element-wise (you don’t simply multiply the
corresponding elements). How is it done?

Row-column multiplication

Multiplying matrices is fairly easy to show, but kind of tough to write a formula for symbolically . The easiest way to
introduce it is to define it in terms of rows and columns.

Suppose you have a row (a 1 × n matrix) and a column (an n × 1 matrix). For example, suppose

We define matrix multiplication by multiplying the row entries of A with the corresponding column entries of B, and
summing the products :

In general, we could write that if

then

• Note than in order for this to work , the number of columns of A must equal the number of rows of B.

Matrix-matrix multiplication

The formal definition of matrix-matrix multiplication:

Given an m × r matrix A and an r × n matrix B, the product matrix C = AB is an m × n matrix
with entries ci,j such that

It is not immediately obvious what that definition is telling you to do, although you can probably see right away
you’re multiplying elements of A and elements of B that correspond in some fashion.

However, having defined row-column multiplication, we can describe the process of matrix-matrix multiplication in
the fol lowing way . What that definition is saying is: to get the i, jth element of the product C = AB, multiply
the ith row of A with the jth column of B

For example, to get c1,1, multiply the first row of A with the first column of B.
To get c2,3, multiply the second row of A with the third row of B.
And so on ...

Sounds complicated? It’s tough to describe in symbols and in words , but makes sense with an example. Go to the
next page and fill in the grid. There’s a screen in the clip that’ll do the calculations .

Example: Given

find the product C = AB.

 Multiplying row 1 of A wit column 1 of B gives c1,1. Multiplying row 1 of A with column 2 of B gives c1,2. Multiplying row 1 of A with column 3 of B gives c1,3. Multiplying row 2 of A with column 1 of B gives c2,1 Multiplying row 2 of A with column 2 of B gives c2,2 Multiplying row 2 of A with column 3 of B gives c2,3, Multiplying row 3 of A with column 1 of B gives c3,1. Multiplying row 3 of A with column 2 of B gives c3,2. Multiplying row 3 of A with column 3 of B gives c3,3.

Observations:
• Do A and B have to be the same size for multiplication to be defined?
• What does have to match up?
• There’s an easy way to remember this (and to also predict the size of the resulting product matrix):

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