10.4 Matrix Algebra

1. Matrix Notation

2. Sum/Difference of 2 matrices

3. Scalar multiple

4. Product of 2 matrices

5. Identity Matrix6. Inverse of a matrix

a) Verify the inverse of a matrix b)Finding the inverse7. Solve a system using inverse matrices

1. Matrix Notation

Notation: refers to the element in row i, column j of a matrix A.

Notation: An “m x n” matrix has m rows and n columns

Example:

Identify the element

ija

23a

2720

854

321

4

5

0

A

2. Sum and Difference of 2 matrices

To add/subtract… add corresponding elements.

1)

2)

13

90A

01

82B

BA

BA

Note: The matrices must be same dimensions!

3. Scalar Multiplication

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

kA implies the number k is multiplied times every element in A:

Example:

Find 1) 2)

13

90A

A2

01

82B

BA 23

4. Matrix MultiplicationMultiplication is NOT like addition (where we added corresponding elements).

You will NOT multiply corresponding elements.

Given:

Find the product:

13112A

3

4

0

1

B

Evaluate

4. Matrix Multiplication

AB

)0)(9()8)(0( AB

)1)(9()2)(0(AB

)1)(1()2)(3(

AB

)0)(1()8)(3(

AB

)0)(1()8)(3()1)(1()2)(3(

)0)(9()8)(0()1)(9()2)(0(AB

13

90A

01

82B

AB

4. Matrix MultiplicationYour turn to practice:

65

72A

73

101B

AB 1)

ABA 3 2)

4. Matrix Multiplication

rows columns rows columns

Example: is not possiblewhen columns in A does notequal rows in B:

nmA

Important: Matrix multiplication can only be performed if

The number of columns in first matrixis equal to

number of rows in second!

pnB

1

6

5

9

4

2

,53

11BA

AB

5. Identity MatrixDefinition: The identity Matrix is a square matrix thathas 1’s on diagonal and 0’s elsewhere

An identity matrix has the same properties as 1 in the real numbers.

10

012I

100

010

001

3I

5. Identity MatrixIdentity Property

Example:

Given the matrix:

AIA

AAI

23

41A

6. Inverse of a Matrix

The Inverse is the matrix A is and satisfies

Example:Given and its inverse

show and

12

13A

IAA 1

1A

32

111A

IAA 1

IAA

IAA

1

1

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

6. b) Finding the Inverse of a Matrix

To 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 |

35

12A

1A

6. b) Finding the Inverse of a Matrix

Example:Find the multiplicative inverse of

Verify when finished!Your turn… Find the inverse for

310

054

111

A

310

054

111

A

7. Solve a system of linear equations using the inverse matrix method

If a system has a unique solution

where A is the coefficient matrix, X and B are 1 column matrices.

then is the solution.

1) Find 2) Multiply

3) The result in 2) is the solution

BAX

BAX 11A

BA 1

X

7. Solve a linear system using inverse Matrix

Example:Solve the system:

Note: We found in an earlier example

23

154

1

zy

yx

zyx

1A

7. Solve a linear system using inverse Matrix

Your turn:Solve the system:

6

532

62

yx

zyx

zx