8.3 Another Way of Solving a System of Equations

Objectives: 1.) Learn to find the inverse matrix

2.) Use the inverse matrix to a system of equations

Consider this

Let A= Y= B=

11

21

2221

1211

aa

aa

22

10

Find Y if A + Y = B

Consider this

Let A= Y= B=

11

21

21

11

a

a

2

0

Find Y if AY = B

There is no division operation on matrices

Alternative Form for Solving a System of Equations Using the Inverse Matrix

New NotationLet A be the cofficient matrix Let X be the variable matrixLet B be the solution matrix

Thus, AX= B

Coefficient Matrix (A)

• A matrix whose real entries are the coefficients from a system of equations

435

243

yx

yx

35

43A

Variable Matrix (X)

• A column matrix of the unknown variables

435

243

yx

yx

y

xX

Solution Matrix

• A column matrix whose entries are the solutions of the system of equations

4

2

435

243

yx

yx

Identity Matrix

• A square matrix with a diagonal of 1s and all other entries are zeros

• RREF Form• Notation: I

Characteristic of the Identity Matrix

• When a matrix is multiplied by the identity, you get the same matrix; AI= A

11

21A

10

01I

Example

11

21A

10

01I

Inverse Matrix

• Let A be a square matrix, then A-1 is the inverse matrix if

AA-1 = I = A-1A

Example

• A = B=

11

21

11

21

Thus B can be notated A-1 because it is the inverse of A.

Finding the Inverse Matrix (The original matrix needs to be square!)

1.) Write the augmented matrix with [A:I] (The coefficient matrix and the identity matrix side by side

2.) Do proper row reductions to both A and I until A is in rref form (It has become an identity matrix itself

3.) The change in I is the inverse matrix of A, A-1

*** If you get a row of full zeros, the inverse does not exist****

Example Pg. 579 #22

Example: Find the inverse matrix of

32

64

How this helps us solve a system of equations. Example: Pg. 580 #53

Shortcut for finding the inverse of a 2x2

• Pg. 577: If

ac

bd

bcadA

11

dc

baA

A is invertible if ad-bc ≠0There is no inverse if ad-bc=0

32

64A A is invertible if ad-bc ≠0

5

4

5

14

3

2

7

A

ac

bd

bcadA

11

Homework: 8.3

• Page 579 # 2; 5; 19-22; 39-47(odd); 53; 54; 60; 71