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Multiply Matrices. Objectives: To multiply matrices. Warm-up. How do you find the dimensions of a matrix? We’ve learned how to add and subtract matrices and how to multiply by a scalar. How do you think we should multiply matrices?. Objective 1. You will be able to multiply matrices.

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multiply matrices
Multiply Matrices

Objectives:

  • To multiply matrices
warm up
Warm-up
  • How do you find the dimensions of a matrix?
  • We’ve learned how to add and subtract matrices and how to multiply by a scalar. How do you think we should multiply matrices?
objective 1
Objective 1

You will be able to multiply matrices

matrix multiplication
Matrix Multiplication

Multiplying two matrices is weird. Thank you, Arthur Cayley (1858).

Because of the way it is defined, if we are multiplying two matrices , then the number of columns in must match the number of rows in .

Arthur Cayley, 1850ish

matrix multiplication1
Matrix Multiplication

If and ,

Must match

Order:

then .

Arthur Cayley, 1850ish

Order:

getting rid of the middle man
Getting Rid of the Middle Man

In general, if matrix has order and has order , then has order .

When multiplying two matrices, the first thing you should do is find the order of the product!

exercise 1
Exercise 1

Can you find ? If so, what is the order of the product?

  • : ; :
  • : ; :
exercise 2a
Exercise 2a

Multiply.

exercise 2b
Exercise 2b

Multiply.

exercise 2c
Exercise 2c

Multiply.

exercise 2d
Exercise 2d

Multiply.

exercise 2d1
Exercise 2d

Multiply.

exercise 2d2
Exercise 2d

Multiply.

exercise 2d3
Exercise 2d

Multiply.

exercise 2d4
Exercise 2d

Multiply.

exercise 2d5
Exercise 2d

Multiply.

matrix multiplication2
Matrix Multiplication

Definition of Matrix Multiplication

This definition will make more sense once we do an application question.

Column

Rowx

matrix multiplication3
Matrix Multiplication

Definition of Matrix Multiplication

Column

+Ro

+ol

+ou

+wm

+wn

= RC

RRooww x

the order of the product
The Order of the Product

So you might be asking why the number of columns of the first matrix must equal the number of rows in the second matrix.

This is because we must multiply a row (in the first matrix) by a column (in the second matrix), and the number of elements must match.

How many elements are there in any row in the first matrix? How many elements are there in any column in the second matrix?

exercise 3a
Exercise 3a

Two softball teams submit equipment lists to their sponsors.

If a bat costs $21, a ball costs $4, and a glove costs $30, use matrices to find the total cost of equipment for each team.

exercise 3b
Exercise 3b

The previous Exercise simplifies to the following matrix multiplication problem. Can you now see why matrix multiplication is so weird?

exercise 4
Exercise 4

, , and

exercise 5
Exercise 5

, , and

properties of matrix multiplication
Properties of Matrix Multiplication

Let , , and be matrices and let be a scalar.

Remember: There is no commutative property for matrix multiplication!

identity matrix
Identity Matrix

Recall that multiplying a number by 1 gives you back the same number. This is the Identity Property of Multiplication.

Similarly, multiplying a matrix by the identity matrix, , will return the original matrix.

identity matrix1
Identity Matrix

The identity matrix has 1s along the main diagonal and zeros everywhere else.

exercise 6
Exercise 6

What is the identity matrix for a ?

3 6 multiply matrices
3.6: Multiply Matrices

Objectives:

  • To multiply matrices