10.4 Matrix Algebra

10.4 Matrix Algebra

1. Matrix NotationA matrix is an array of numbers.

Definition: The Dimension of a matrix is m x n “m by n” where m = # rows, n = #columns

41

32

03

A

2. Sum and Difference of 2 matrices

To add/subtract… add corresponding elements.

Evaluate:

1390

A

0182

B

BA

Note: The matrices must be same dimensions!

3. Scalar Multiplication

We can multiply matrix by a number (known as scalar).

Example:

Find

1390

A

A2

4. Matrix MultiplicationMultiplication : Row-by-Column multiplication

Determine

dcba

A

hgfe

B

AB

Evaluate

4. Matrix Multiplication

AB

1390

A

0182

B

AB

4. Matrix Multiplicationmore practice:

6572

A

73101

B

AB 1)

ABA 3 2)

4. Matrix Multiplication

Dimensions: rows columns rows columns

AB will have dimensions

nmA

Important: For Matrix multiplication to work:

The number of columns in first matrixmust equal

number of rows in second!

pnB

AB Determine

4321

A

51

31

20

B

pm

Why is the product BA not possible?

4. Matrix Multiplication

Evaluate the following:

31

10

21C

51

31

20

B

BC 1)

CB 2)

100010001

D

BD 3)

5. Identity MatrixReal Numbers: 1 is the multiplicative identity. Example

Matrices: is the Multiplicative identity of a matrix ,

a square matrix with 1’s on diagonal, 0’s elsewhere.

is used to represent the order n (dimension)

Example: Order 2 Order 3

A matrix times its identity returns the original matrix.

1001

2I

100010001

3I

nI

AAI

aa 1

I

6. Inverse of a MatrixReal Numbers: Multiplicative Inverse of is (for any )

Matrices: Multiplicative Inverse of a matrix is a matrix read as: “A-inverse”

with the property:

1A

IAA

IAA

1

1

Definition:If a matrix does not have an inverse, it is called singular

a a/10a

11

aa

A

6. Inverse of a Matrix

Example:

Given and its inverse

show and

1213

A

IAA 1

32111A

IAA 1

IAA

IAA

1

1

6. b) Finding the Inverse of a MatrixTo find the inverse:

1) Form augmented matrix

2) Transform to reduced row echelon form (Gauss-Jordan).

3) The identity matrix will magically appear on the right hand side of the bar! This is

1A

Example:Find the multiplicative inverse of

Verify it when finished!

IA |

3512

A

1A

6. b) Finding the Inverse of a Matrix

Example:Find the multiplicative inverse of

Graphing calculator: To Enter Matrix data:

• 2nd MATRIX: Edit (Enter)• Dimensions 3 x 3

To find Inverse:• 2nd MATRIX: NAMES 1:[A] Enter “^-1”.

310054111

A

7. Solve a system of linear equations Inverse Matrix method

A system can be written using matrix notation:

A is the coefficient matrixB is the constant matrixX represents the unknowns.

Example: Write this system using matrix notation:

BAX

231541

zyyxzyx

7. Inverse matrix method

If has a unique solution

then is the solution.

Solve:

BAX

BAX 1

231541

zyyxzyx

7. Solve a linear system using inverse Matrix

Example:Solve the system:

Note: We found in an earlier example

231541

zyyxzyx

1A

7. Solve a linear system using inverse Matrix

Your turn:Solve the system:

6532

62

yxzyx

zx