PRECALCULUS 2

Determinants, Inverse Matrices & Solving

December

• DO NOW:

TAKE OUT PAPER OR A NOTEBOOK!!!!

CW: INVERSE POWER POINT!!!

HW: INVERSE wksts.

• SWBAT:

• Identify the inverse of a 2x2 matrix

• Identify the inverse of a 3 x 3 matrix

45

23

Notice the different symbol:

the straight lines tell you to

find the determinant!!

(3 * 4) - (-5 * 2)

12 - (-10) 22

=45

23

Finding Determinants of Matrices

=

=

241

521

302

2

1

-1

0

-2

4

= [(2)(-2)(2) + (0)(5)(-1) + (3)(1)(4)] [(3)(-2)(-1) + (2)(5)(4) + (0)(1)

(2)][-8 + 0 +12]

-

- [6 + 40 + 0]

4 – 6 - 40

Finding Determinants of Matrices

=

= = -42

Inverse Matrix:

Using matrix equations

2 x 2

dc

ba

In words:•Take the original matrix. •Switch a and d. •Change the signs of b and c. •Multiply the new matrix by 1 over the determinant of the original matrix.

ac

bd

bcad

1 1A

A

24

410

)4)(4()10)(2(1

24

410

41

=

21

1

125

Using matrix equations

Example: Find the inverse of A.

104

42A

1A

1A

Find the inverse matrix.

25

38

Det A = 8(2) – (-5)(-3) = 16 – 15 = 1

Matrix A

Inverse =

det

1 MatrixReloaded

85

3211

= =

85

32

10

01

Identity matrix: Square matrix with 1’s on the diagonal and zeros everywhere else

2 x 2 identity matrix

100

010

001

3 x 3 identity matrix

The identity matrix is to matrix multiplication as ___ is to regular multiplication!!!!1

Using matrix equations

Multiply:

10

01

43

25=

43

25

10

01

43

25=

43

25

So, the identity matrix multiplied by any matrix lets the “any” matrix keep its identity!

Mathematically, IA = A and AI = A !!

What happens when you multiply a matrix by its inverse?

1st: What happens when you multiply a number by its inverse?71

7

A & B are inverses. Multiply them.

85

32=

25

38

10

01

So, AA-1 = I

Why do we need to know all this?To Solve Problems!Solve for Matrix X.

=

25

38X

13

14

We need to “undo” the coefficient matrix. Multiply it by its INVERSE!

85

32=

25

38X

85

32

13

14

10

01X =

34

11

X =

34

11

You can take a system of equations and write it with

matrices!!!

3x + 2y = 11

2x + y = 8becomes

12

23

y

x=

8

11

Coefficient

matrix

Variable

matrix

Answer matrix

Using matrix equations

Let A be the coefficient matrix.

Multiply both sides of the equation by the inverse of A.

8

11

8

11

8

11

1

11

Ay

x

Ay

xAA

y

xA

12

23 -1=

32

21

11

=

32

21

32

21

12

23

y

x=

32

21

8

11

10

01

y

x=

2

5

y

x=

2

5

Using matrix equations

12

23

y

x=

8

11Example: Solve for x and y .

1A

Wow!!!!

3x + 2y = 11

2x + y = 8

x = 5; y = -2

3(5) + 2(-2) = 11

2(5) + (-2) = 8

It works!!!!

Using matrix equations

Check:

You Try…

Solve:

4x + 6y = 142x – 5y = -9

(1/2, 2)