Determinants & Inverse Matrices

Determinants & Inverse Matrices

The

of the 2 2 matrix

determinant

ab

cd

is the number

.

ad cb

The above sentence is abbreviated as

det a b = ad cb

cd

Example.

det

4 1

2 3

= 4(

3)

1( 2) = 12 + 2 = 10

The determinant of a 3 3 matrix can be found using the formula

0

1

ab c

det @

A = det e f

det d f + det d e

def a

b

c

hi

gi

gh

gh i

Example.

0 21

det @0 3 10

0 2 1

1

A = 2 det

3 0

2 1

(

1) det

0 1

2 1

+ 0 det

0 1

3 0

= 2[3 ? 1 0( 2)] + [0 ? 1 1( 2)] + 0

=2?3+2

=8

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

286

Determinants and inverses

A matrix has an inverse exactly when its determinant is not equal to 0.

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

22 inverses

Suppose that the determinant of the 2 2 matrix

ab

cd

does not equal 0. Then the matrix has an inverse, and it can be found using the formula

1

ab =

1

d

b

cd

det a b

ca

cd

Notice that in the above formula we are allowed to divide by the determinant since we are assuming that it's not 0.

Example. To find

35 1 12

first check that

det

3 1

5 2

=3?2

1?5=1

Then

35 12

1=1 1

2 1

5 3

=

2 1

5 3

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

287

33 inverses

There is a way to find an inverse of a 3 3 matrix ? or for that matter, an matrix ? whose determinant is not 0, but it isn't quite as simple as

nn finding the inverse of a 2 2 matrix. You can learn how to do it if you take a linear algebra course. You could also find websites that will invert matrices for you, and some calculators can find the inverses of matrices as long as the matrices are not too large.

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

288

Exercises

For #1-6, compute the determinant of the given matrix.

1.)

21

11

2.)

1

01

3.)

42

21

4.)

0

1

3 00

@1p07 1 0A 2 26

5.)

0

1

111

@2 2 2A

333

6.)

0

1

121

@2 1 2A

134

7.) Which of the six matrices from the previous problems have inverses?

Find the inverses of the matrices below.

8) .

10 51

9) .

35 23

289

10 ) .

42 13

I

/

a

Match the functions with their graphs. (

11.) ( ) = 2 fx x

13.) ( ) = x px

2

x

if 2 ( 1 0);

x

,

if

x

2

[0 ,

1).

I 12.) ( ) = gx x

J7, i (

2

14.)

() qx

=I

x

r

if 2 ( x

1 ,

02);

if 2 [0 1).

xx,

1.

A.)

B.)

I / II I

\

4--'

a

--2

I I

C.)

290

II // D.)

i

7,

27

285

\

I

4--'

290

i i --2

I

4--'

 ................
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