IXL - Properties of matrices (Precalculus practice)

ref_doc_title.

or

Learn with an example

question

If

E

=

–

2

3

6

–

2

and

F

=

–

4

2

4

–

4

,

is the following statement true or false?

E

F

=

F

E

true

false

key idea

A matrix product exists if and only if the number of columns in the first matrix equals the number of rows in the second matrix. If

A

has dimensions

m

×

n

and

W

has dimensions

n

×

p

,

then

A

W

has dimensions

m

×

p

.

A

W

=

A

W

(

m

×

n

)

(

n

×

p

)

m

×

p

Two matrices are equal if and only if their corresponding elements are equal.

solution

To see if the statement is true or false, find

E

F

and

F

E

.

Since

E

=

–

2

3

6

–

2

and

F

=

–

4

2

4

–

4

are square matrices with the same number of rows and columns, you can multiply them in either order.

E

F

and

F

E

will also be square matrices with 2 rows and 2 columns.

Since both matrix products exist, you can calculate them.

Calculate

E

F

.

Find the element in the

1

^st

row and

1

^st

column of

E

F

.

Multiply each element in the

1

^st

row of

E

by the corresponding element in the

1

^st

column of

F

.

Then, add the products.

E

F

=

–

2

3

6

–

2

–

4

2

4

–

4

=

–

2

(

–

4

)

+

3

(

4

)

?

=

8

+

12

?

=

20

?

Apply a similar process to find the element in the

1

^st

row and

2

^nd

column of

E

F

.

–

2

3

6

–

2

–

4

2

4

–

4

=

20

–

2

(

2

)

+

3

(

–

4

)

?

=

20

–

4

–

12

?

=

20

–

16

?

Repeat this process to calculate the remaining elements of

E

F

.

–

2

3

6

–

2

–

4

2

4

–

4

=

20

–

16

–

32

20

Calculate

F

E

.

Find the element in the

1

^st

row and

1

^st

column of

F

E

.

Multiply each element in the

1

^st

row of

F

by the corresponding element in the

1

^st

column of

E

.

Then, add the products.

F

E

=

–

4

2

4

–

4

–

2

3

6

–

2

=

–

4

(

–

2

)

+

2

(

6

)

?

=

8

+

12

?

=

20

?

Apply a similar process to find the element in the

1

^st

row and

2

^nd

column of

F

E

.

–

4

2

4

–

4

–

2

3

6

–

2

=

20

–

4

(

3

)

+

2

(

–

2

)

?

=

20

–

12

–

4

?

=

20

–

16

?

Repeat this process to calculate the remaining elements of

F

E

.

–

4

2

4

–

4

–

2

3

6

–

2

=

20

–

16

–

32

20

So,

E

F

=

20

–

16

–

32

20

and

F

E

=

20

–

16

–

32

20

.

Compare

E

F

and

F

E

.

Since

E

F

=

F

E

,

the statement is true.

However, this statement is not true for all matrices. In general, matrices do not satisfy the commutative property of multiplication. The statement

E

F

=

F

E

is true in this case only because of the particular values of

E

and

F

.

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