Multiplying MatricesDr. ShildneckFall, 2015

Can You Multiply Matrices?› What do you think has to be true in order to

multiply?

› What procedure do you think we’d use?

› Let’s try this one…

[1 24 2][ 2 3

−1 5]

When Can You Multiply Matrices?

› Was your answer… ?

Then you are WRONG!

› The correct answer is…

[ 2 6−4 10]

[0 136 22]

INVESTIGATIONTake the Handout and Read over SECTION ONE.

Complete the Front Page with you calculator.

INVESTIGATIONThe answers to the questions for SECTION ONE are…

6)

4) 3)

2) 1)

5) Not Possible

So, How Do We Multiply Matrices?› First of all, matrices can only be multiplied if the

dimensions “match up” correctly… this does not mean they have to have the same dimensions.

› Use your calculator to complete the problems in SECTION TWO of the hand out. Make a conjecture about WHEN matrices can be multiplied.

Hint: You might want to examine the dimensions of the matrices you are trying to multiply.

So, How Do We Multiply Matrices?› What ideas did you come up with regarding WHEN

we CAN multiply matrices?

Two matrices can be multiplied if and only if the

number of ______________ in the first one is EQUAL to

the number of ______________ in the second one.

COLUMNS

ROWS

So, How Do We Multiply Matrices?› The easiest way to do a quick check to see if you

can multiply is to write the dimensions of each down in order.

› If the middle numbers match, you CAN multiply.

Let’s look further…

› Looking back at the problems in Section 2 that could be multiplied…

› Write down the dimensions of each matrix in the problem› Write down the dimensions of the answer to each problem› Make a conjecture regarding how the dimensions in the problem

relate to the dimensions of the answer.

So, How Do We Multiply Matrices?› What ideas did you come up with regarding the

dimensions of the product (answer)?

The dimensions of the product of two matrices is the

number of ______________ in the first by the number of

______________ in the second.

› In other words, the dimensions are the “outside” numbers

COLUMNS

ROWS

So, now, how do we find the entries?› The reason the columns and rows must match up is

the process used to determine the entries...

› I call this process “The Finger Method”

Looking back at our original problem…

[1 24 2][ 2 3

−1 5]

[1 24 2][ 2 3

−1 5]First, check the dimensions…

2 x 2 2 x 2They match… so we can multiply!

Next, what will the dimensions of the answer be?

The dimensions of the answer will be 2x2.Now draw a BLANK matrix with

the correct dimensions for the answer…

[ ]¿¿¿¿

[1 24 2][ 2 3

−1 5]

[ ]¿¿¿¿

1(2)

+ 2(-1)

To find the entry for the FIRST ROW , FIRST COLUMN use the FIRST ROW and FIRST COLUMN(Remember to always start at the beginning of the row and top of the column)

To find the entry for the FIRST ROW , 2nd COLUMN use the FIRST ROW and 2nd COLUMN(Remember to always start at the beginning of the row and top of the column)

1(3)

+ 2(5)

To find the entry for the 2nd ROW , 1st COLUMN use the 2nd ROW and 1st COLUMN(Remember to always start at the beginning of the row and top of the column)

4(2)

+ 2(-1)

To find the entry for the 2nd ROW , 2nd COLUMN use the 2nd ROW and 2nd COLUMN(Remember to always start at the beginning of the row and top of the column)

4(3)

+ 2(5)

[1 24 2][ 2 3

−1 5]

[ ]¿¿¿¿

1(2)

+ 2(-1)

1(3)

+ 2(5)

4(2)

+ 2(-1)

4(3)

+ 2(5)

[ ]¿¿¿¿

0 136 22

=

=

The Associative Property

The Associative Property for Multiplication states

(A x B) x C= A x (B x C)

For Matrices, this is also true…

This means that you can multiply either set first and get the same answer…

Given that we just found…

[1 24 2][ 2 3

−1 5] =

What do you think

[1 24 2][ 2 3

−1 5]

equals?

The Commutative Property

The Commutative Property for Multiplication states

A x B = B x A

But, in this last example we found that it does not hold true.

In fact, it is rare for matrices to be commutative with respect to multiplication…

The Commutative Property

Since the Commutative Property for Multiplication does not hold true for Matrices, it is extremely important that

you remember…

NEVER SWITCH THE ORDER OF MATRICES

WHEN MULTIPLYING!!!

More Examples for Multiplication

[1 0 42 4 2] [0 1

2 35 2]

EXAMPLE 2

More Examples for Multiplication

[1 −13 2 ] [1 0

0 23 4 ]

EXAMPLE 3

More Examples for Multiplication

[123] [4 2 5 ]

EXAMPLE 4

More Examples for Multiplication

[1 2 3 ] [251]EXAMPLE 5

More Examples for Multiplication

[1 3 02 5 3 ] [1 2 5

1 3 2]EXAMPLE 6

More Examples for Multiplication

[1 3 02 𝑥+1 3 ] [ 12𝑥4 ]

EXAMPLE 7

More Examples for Multiplication

[1 3 02 5 3 ] [1 2

2 03 5 ] [12 ]

EXAMPLE 8

ASSIGNMENTAssignment #5 Worksheet