3⇥ 3 MatricesMuch of this chapter is similar to the chapter on 2⇥ 2 matrices. The most

substantial di↵erence between 2 ⇥ 2 matrices and 3 ⇥ 3 matrices is that it’sharder to write a 3⇥ 3 matrix than it is to write a 2⇥ 2 matrix.

3 ⇥ 3 matrices have 3 rows and 3 columns. They are a square block of 9numbers, such as 0

@2 0 64 �5 14

�10 3 4

1

A

Two matrices are equal if the entry in any position of the one matrix equalsthe entry in the same position of the other matrix.

Examples.0

@3 2 75 0 �16 9 4

1

A =

0

@3 2 75 0 �16 9 4

1

A but

0

@3 2 75 0 �16 9 4

1

A 6=

0

@3 2 75 9 �16 0 4

1

A

* * * * * * * * * * * * *

Matrices as functionsA 3⇥ 3 matrix defines a function whose domain is R3 and whose target is

R3. The function is defined as follows:0

@a b c

d e f

g h i

1

A

0

@u

v

w

1

A =

0

@au+ bv + cw

du+ ev + fw

gu+ hv + iw

1

A

Notice that the first, second, or third entry in the vector on the right sideof the above equation can be found by multiplying the first, second, or thirdrow of the matrix

0

@a b c

d e f

g h i

1

A

and the column278

0

@u

v

w

1

A

Example. The matrix 0

@5 3 1

�2 2 47 0 �1

1

A

has 3 rows and 3 columns, so it is a function whose domain is R3, and whosetarget is R3.Because, 0

@29

�3

1

A

is a vector in R3, 0

@5 3 1

�2 2 47 0 �1

1

A

0

@29

�3

1

A

is also a vector in R3. The vector it equals is

0

@5 3 1

�2 2 47 0 �1

1

A

0

@29

�3

1

A =

0

@5 · 2 + 3 · 9 + 1 · (�3)�2 · 2 + 2 · 9 + 4 · (�3)7 · 2 + 0 · 9 + (�1) · (�3)

1

A

=

0

@10 + 27� 3�4 + 18� 1214 + 0 + 3

1

A

=

0

@34217

1

A

Identity matrixNotice that0

@1 0 00 1 00 0 1

1

A

0

@x

y

z

1

A =

0

@1 · x+ 0 · y + 0 · z0 · x+ 1 · y + 0 · z0 · x+ 0 · y + 1 · z

1

A =

0

@x

y

z

1

A

279

Thus, 0

@1 0 00 1 00 0 1

1

A

is the identity function whose domain is R3. We call this matrix the 3 ⇥ 3identity matrix.

* * * * * * * * * * * * *

Matrix multiplicationYou can “multiply” two 3⇥ 3 matrices to obtain another 3⇥ 3 matrix.

Order the columns of a matrix from left to right, so that the 1st columnis on the left, the 2nd column is directly to the right of the 1st, and the 3rd

column is to the right of the 2nd.To multiply two matrices, call the columns of the matrix on the right “input

columns”, and put each of the input columns into the matrix on the left(thinking of it as a function). The column that is assigned to the 1st inputcolumn by the matrix function will be the 1st column of the product you aretrying to find.The column that is assigned to the 2nd input column by the matrix function

will be the 2nd column of the product, and the column that is assigned tothe 3rd input column by the matrix function will be the 3rd column of theproduct.

Example. To find the product0

@2 7 31 5 80 4 1

1

A

0

@3 0 12 1 01 2 4

1

A

separate the matrix on the right into its three “input columns”:0

@3 0 12 1 01 2 4

1

A 7!

0

@321

1

A

0

@012

1

A

0

@104

1

A

280

Then the product0

@2 7 31 5 80 4 1

1

A

0

@3 0 12 1 01 2 4

1

A

will be a 3⇥ 3 matrix whose first column (when read left-to-right) equals0

@2 7 31 5 80 4 1

1

A

0

@321

1

A =

0

@23219

1

A

whose second column equals0

@2 7 31 5 80 4 1

1

A

0

@012

1

A =

0

@13216

1

A

and whose third column is0

@2 7 31 5 80 4 1

1

A

0

@104

1

A =

0

@14334

1

A

To repeat the previous sentence,0

@2 7 31 5 80 4 1

1

A

0

@3 0 12 1 01 2 4

1

A =

0

@23 13 1421 21 339 6 4

1

A

Matrix multiplication is function composition

If A and B are 3⇥ 3 matrices, then the result of multiplying the matricesAB would determine the same function AB : R3 ! R3 as the function thatresults from composition, namely A �B.

* * * * * * * * * * * * *

281

Inverse matricesIf B is a 3⇥ 3 matrix, then B

�1 is the 3⇥ 3 matrix where

BB

�1 =

0

@1 0 00 1 00 0 1

1

A

and

B

�1

B =

0

@1 0 00 1 00 0 1

1

A

Example. We can check that the two matrices0

@7 0 24 1 13 0 1

1

A and

0

@1 0 �2

�1 1 1�3 0 7

1

A

are inverses of each other by multiplying them in either order and checkingto see that their product is the identity matrix:

0

@7 0 24 1 13 0 1

1

A

0

@1 0 �2

�1 1 1�3 0 7

1

A =

0

@1 0 00 1 00 0 1

1

A

and0

@1 0 �2

�1 1 1�3 0 7

1

A

0

@7 0 24 1 13 0 1

1

A =

0

@1 0 00 1 00 0 1

1

A

* * * * * * * * * * * * *

282

Exercises

1.) The matrix

A =

0

@1 1 10 2 �33 �2 0

1

A

describes a function A : R3 ! R3.Find the vectors0

@1 1 10 2 �33 �2 0

1

A

0

@014

1

A and

0

@1 1 10 2 �33 �2 0

1

A

0

@�210

1

A

2.) The matrix

B =

0

@3 2 �21 0 104 �5 7

1

A

describes a function B : R3 ! R3.Find the vectors0

@3 2 �21 0 104 �5 7

1

A

0

@12

�1

1

A and

0

@3 2 �21 0 104 �5 7

1

A

0

@302

1

A

3.) Find the product0

@0 1 02 0 20 3 0

1

A

0

@2 �1 50 3 10 0 �4

1

A

4.) Find the product0

@2 �1 50 3 10 0 �4

1

A

0

@0 1 02 0 20 3 0

1

A

283

5.) Find the product0

@3 �17 517 3 3441 3 18

1

A

0

@1 0 00 1 00 0 1

1

A

For #6 and #7, determine whether the two matrices given are inverses ofeach other.

6.)

0

@4 1 21 0 11 �1 0

1

A and

0

@0 2 11 1 3

�1 0 2

1

A

7.)

0

@�1 1 0�2 1 10 0 1

1

A and

0

@1 �1 12 �1 10 0 1

1

A

284

Match the functions with their graphs.

8.) (x+ 1)2 + 2 10.) (x� 1)2 + 2

9.) (x+ 1)2 � 2 11.) (x� 1)2 � 2

A.) B.)

C.) D.)

285

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