Have we ever seen this phenomenon before?

130

21312

185

100

010

001

130

21312

185

Let’s do some quick multiplication…

130

21312

185

100

010

001

130

21312

185

What do we call this?

5 • __= 5the

(multiplicative) identity

1

Let’s do some quick multiplication…

130

21312

185

100

010

001

130

21312

185

What will we call this?

Identity Matrix

I

Let’s do some quick multiplication…

Identity MatrixIdentity Matrix

10

01I

2x2 Identity Matrix

100

010

001

I

3x3 Identity Matrix

Identity MatrixIdentity Matrix

AI= AIA= A

Any matrix multiplied by the identity matrix will be equal to itself.

Identity MatrixIdentity Matrix

10

01

10

01

Or

63

32

63

32

63

32

63

32

Inverse MatrixInverse Matrix

A * A-1= I

Any matrix multiplied by its inverse will be the identity matrix.

Solving Systems using Solving Systems using MatricesMatrices

Matrix EquationMatrix Equation

6

8

21

45

y

x

A linear system can be written as a matrix equation AX=B

Coefficient matrix Variable

matrix

Answer matrix

5 4 8

1 2 6

x y

x y

Ex. 1Ex. 1 Write as a matrix equation. Write as a matrix equation.

3 4 5

2 10

x y

x y

Suppose ax = b

How do you solve for x?

We cannot divide matrices, but we can multiply by the inverse.

AX = BAA-1-1 AA-1-1

IX = AA-1-1B

X = AA-1-1B

Solving Matrix EquationsSolving Matrix Equations

Ex. 2Ex. 2 Solve using matrices. Solve using matrices.

3 4 5

2 10

x y

x y

x = -7

y = -4

10

5

12

43

y

x

A B

X = AA-1-1B

AX = B

(-7, -4)

Ex. 3Ex. 3 Solve using matrices Solve using matrices7 3 11

14 4 2

x y

x y

x = 5/7

y = 2

(5/7, 2)

Ex. 4Ex. 4 Solve using matrices Solve using matrices

2 3 1

3 3 1

2 4 2

x y z

x y z

x y z

x = 2

y = -1

z = -2

(2, -1, -2)

Ex. 5Ex. 5 Solve using matrices Solve using matrices

x y z

x y

x y z

3 5 15

2 1

9 8 4 12

x = 4

y = -7

z = 2

(4, -7, 2)