Post on 04-Jan-2016
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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