Chapter 8Matrices and Determinants

By Richard Warner, Nate Huyser, Anastasia Sanderson, Bailey Grote

Chapter 8.1: General Matrices

• Rectangular array of numbers called entries • Dimensions of a matrix are number of rows by

the number of columns

333231

232221

131211

AAA

AAA

AAA

Chapter 8.1: Augmented Matrices

• Augmented Matrix- derived from a system of equations

• Elementary Row Operations• Interchange any two rows• Multiply any row by a nonzero constant• Add two rows together

• 2x2 by hand, 3x3 with calculator

Chapter 8.1: Reduced Row Echelon Form (RREF)

• Any rows consisting of all zeros occur at the bottom of the matrix

• All entries on the main diagonal are 1• All entries not on the main diagonal or in the last

column are 0• A13 is the x-coordinate of the solution

• A23 is the y-coordinate of the solution

y

x

10

01

Chapter 8.1: Gauss Jordan Elimination

• Uses Augmented Matrices to solve systems of equations

1. Write system as an augmented matrix2. Use the row operations to make A11 = 1

3. Work down, around, and up to achieve RREF4. Write last column as ordered pair for final answer

Chapter 8.1: Solving with Calculator (RREF)

• Only used for Matrices larger than 2x21. (2nd) [Matrix] → EDIT2. Matrix[A] 3x4 3. Enter entries by rows4. (2nd) [Quit]5. (2nd) [Matrix] → MATH6. Select [RREF] 7. (2nd) [Matrix] select Martix[A]

z

y

x

100

010

001

Chapter 8.2: Matrix Operations

Equality of Matrices: 2 matrices are equal if they have the same dimensions and their corresponding entries are equal

To add and subtract Matrices: They must have the same dimensions.•Add the corresponding entries

Scalar Multiplication: •Multiplying a matrix by a scalar (constant)•Multiply each entry in the matrix by the scalar

Chapter 8.2: Matrix Operations

Matrix Multiplication:• To Multiply AB, A’s columns must equal B’s

rows• Multiply the entries in A’s rows by the

corresponding entries in B’s columns• Amxn* Bnxr =ABmxr

Ex: p.598 #29

Identity Matrices

10

01I 2x2 I 3x3

100

010

001

8.3 Inverse Matrices

• A A-1 =A-1 A =I• If A= where ad-bc cannot equal 0,

Then A-1 =1/(ad-bc) *

dc

ba

ac

bd

Inverse of2x2:

Cont.

8.3 Inverse Multiplication

Inverse of 3x31. Enter [matrix] in calculator2. [matrix][A] [enter] [x-1 ] [enter]To solve a system of linear equations3. Write the system of equations as a matrix problem

4. Find A-1 5. X=A1B

333

222

111

zyx

zyx

zyx

z

y

xx

3

2

1

c

c

c=

Cont.

8.3 Inverse Multiplication

8.4 Determinants

• a real number derived from a square matrix• If A = then Det[A]= AD-CB• For 2x2 matrices only

• For 3x3 matrices or larger1. (2nd) Matrix → [Edit] A2. Enter dimensions3. (2nd) Quit4. (2nd) Matrix → [Math] enter5. (2nd) Matrix → enter

DC

BA

8.5 Determinant Applications

• Cramer’s Rule solves systems using determinates.

• Example:DD

xx

D

Dy

y

dc

ba

df

be

x

dc

ba

fc

ea

y

8.5 Determinant Applications

• Finding the area of a triangle where the points are (a,b), (c,d), (e,f)

• Points are collinear if A=0

A

fe

dc

ba

1

1

1

2

1

0

1

1

1

fe

dc

ba