Matrix Multiplication

How to multiply a 1x3 row by a 3x1 column with the row on the left.

Similarly  we can multiply a 1xn row by a nx1 column.

Examples:

We can multiply any mx3 matrix by a 3x1 column by multiplying each row of the  mx3 by the 3x1 column. Row 1 of the mx3 multiplied by the column gives Row 1 of the product. Row 2 of the mx3 multiplied by the column gives Row 2 of the produc t.

Example:    =

Notice the result of multiplying the 2x3 by the 3x1 is a 2x1 matrix.

We can multiply any mx3 matrix, A, by any 3xn matrix, B, if A is on the left side of B.

You will see that AB need not equal BA, even if both can be done. The product of an mx3 and a 3xn is an mxn matrix. Row i of the product is the result of multiplying each row i of A by each column of B.

In short, If AB=C then

Example:

Since A is 2x3 and B is 3x4, AB can be done and is 2x4.

If we just want the entry in row 2, column 3 of AB we multiply row 2 of A by column 3 of B to get

0(-2)+2(-5)+4(3)=0-10+12 = 2

The whole product is

We can find this in the calculator as follows:  Enter matrix A and enter matrix B.

MATRIX select A, enter, MATRIX select B, enter, enter.

Meanings of Matrix Products

Example

An upholsterer has an order to cover 4 chairs and 3 sofas. Each chair requires 5 yards of fabric and 10 hours of labor. Each sofa requires 16 yards of fabric and 24 hours of lab or. Fabric costs \$18 per yard, labor costs \$20 per hour. Write a matrix product that gives the total cost of the order.

I. Put the information about fabric and labor requirements for chairs and sofas into a 2x2 matrix.

Label the rows and columns so you know what the entries mean. This is also important because we have two choices for setting up the matrix.

Choice 1             Choice 2

The 2nd matrix is the transpose of the first. The problem can be worked with either one so I will use choice 1 first and choice 2 2nd. Where you put the following matrices and how the product looks depends on which choice  you use.

II. Since we have chosen choice 1, the information about fabric cost per yard and labor cost per hour must go on the left as a 1x2 row.

Notice how the labels match up.

III. Put the information  about the number of chairs and sofas ordered in a 2x1 column on the right.

Again, observe how the labeling works.

Making choice 2 it looks like:

The product is \$3464 either way.