Determinants, Inverse Matrices & Solving

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Determinants, Inverse Matrice & Solving

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Determinants, Inverse Matrices & Solving. Finding Determinants of Matrices. Notice the different symbol: the straight lines tell you to find the determinant!!. -. (-5 * 2). =. (3 * 4). 12 - (-10). =. 22. =. 2. 0. 1. -2. -1. 4. - PowerPoint PPT Presentation

Transcript of Determinants, Inverse Matrices & Solving

Page 1: Determinants, Inverse Matrices  & Solving

Determinants, Inverse Matrices & Solving

Page 2: Determinants, Inverse Matrices  & Solving

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

=

=

Page 3: Determinants, Inverse Matrices  & Solving

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

Page 4: Determinants, Inverse Matrices  & Solving

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

Page 5: Determinants, Inverse Matrices  & Solving

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 !!

Page 6: Determinants, Inverse Matrices  & Solving

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

Page 7: Determinants, Inverse Matrices  & Solving

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

Page 8: Determinants, Inverse Matrices  & Solving

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

Page 9: Determinants, Inverse Matrices  & Solving

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

Page 10: Determinants, Inverse Matrices  & Solving

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

Page 11: Determinants, Inverse Matrices  & Solving

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

Page 12: Determinants, Inverse Matrices  & Solving

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

Page 13: Determinants, Inverse Matrices  & Solving

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:

Page 14: Determinants, Inverse Matrices  & Solving

You Try…

Solve:

4x + 6y = 142x – 5y = -9

(1/2, 2)

Page 15: Determinants, Inverse Matrices  & Solving

You Try…

Solve: 2x + 3y + z = -13x + 3y + z = 12x + 4y + z = -2

(2, -1, -2)

Page 16: Determinants, Inverse Matrices  & Solving

Real Life Example:

You have $10,000 to invest. You want to invest the money in a stock mutual fund, a bond mutual fund, and a money market fund. The expected annual returns for these funds are given in the table.You want your investment to obtain an overall annual return of 8%. A financial planner recommends that you invest the same amount in stocks as in bonds and the money market combined. How much should you invest in each fund?

Page 17: Determinants, Inverse Matrices  & Solving
Page 18: Determinants, Inverse Matrices  & Solving

To isolate the variable matrix, RIGHT multiply by the inverse of A

1 1A AX A B 1X A B

Solution: ( 5000, 2500, 2500)