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

=

=

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

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 !!

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

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)

You Try…

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

(2, -1, -2)

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?

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

1 1A AX A B 1X A B

Solution: ( 5000, 2500, 2500)