No Slide Title
Download
Report
Transcript No Slide Title
Solving Linear Systems of Equations - Inverse Matrix
• Consider the following
system of equations ...
a1x b1 y c1
• Let the matrix A
represent the coefficients ...
a1
A
a 2
a 2 x b2 y c2
• Let matrix B hold the
constants ...
b1
b2
c1
B
c 2
x
X
y
• Finally, let matrix X
represent the variables ...
Table of Contents
Solving Linear Systems of Equations - Inverse Matrix
• Now notice what the result is when we work out the
following matrix equation ...
a1
AX B
a 2
b1 x c1
b 2 y c 2
a1x b1 y c1
a 2 x b 2 y c 2
a1x b1 y c1
a 2 x b2 y c2
Table of Contents
Slide 2
Solving Linear Systems of Equations - Inverse Matrix
• Thus, AX = B represents the system of equations.
This matrix equation can be solved for X as follows ...
• Recall that matrix multiplication
is not commutative, so each side of
the equation must be multiplied on
the left by A-1
• Matrix multiplication is
associative.
AX B
1
1
A (AX) A B
1
1
( A A) X A B
1
( I) X A B
1
XA B
Table of Contents
Slide 3
Solving Linear Systems of Equations - Inverse Matrix
• Method of solution:
(1) Given a system of equations, form matrices
A, X, and B.
A
Coefficients
X
Variables (vertical matrix)
B
Constants (vertical matrix)
(2) Find A-1.
(3) Find the solution by multiplying A-1 times B.
X = A-1 B
Table of Contents
Slide 4
Solving Linear Systems of Equations - Inverse Matrix
• Example:
Use an inverse matrix to solve the
system at the right.
3 2
A
1 1
x
X
y
3x 2 y 8
x y3
8
B
3
• Using the methods of finding an inverse, A-1 is ...
1 2
A
1
3
1
Table of Contents
Slide 5
Solving Linear Systems of Equations - Inverse Matrix
• Now find X ...
1 2 8
2
XA B
1 3 3
1
1
• The solution is (2, -1), or
x=2
y = -1
Table of Contents
Slide 6
Solving Linear Systems of Equations - Inverse Matrix
• This same method can be used on any size system of
equations as long as the coefficient matrix is square and
the solution is unique.
Table of Contents
Slide 7
Solving Linear Systems of Equations - Inverse Matrix
Table of Contents