Cramer's Method for Solving Systems of Linear Equations, Slides of Computational Methods

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Chp8 Linear

Algebraic Eqns-

Learning Goals

• Define Linear Algebraic Equations
• Solve Systems of Linear Equation by Hand

using

• Gaussian Elimination
• Cramer’s Method
• Distinguish between Equation System

Conditions: Exactly Determined,

Overdetermined, Underdetermined

• Use MATLAB to Solve Systems of Eqns

Cramer’s Method – Illustrated-

• Cramer’s Method can Solve “Square” Systems; - i.e., [No. Eqns] = [No. Unknowns]
• Consider Sq Sys

8 4 22 50

3 6 2 3

21 9 12 33

− − + =

− + − =

− − = −

x y z

x y z

x y z

 Calc Cramer’s Determinant, Dc

• Also Called the “Characteristic” or “Denominator” Determinant  Dc ≡ Determinant of the Coefficients

8 4 22

3 6 2

21 9 12

− −

− −

− − Dc =

Cramer’s Method – Illustrated-

• Now, to Find The Individual Solns, Sub The Constraint Vector for the Variable Coefficients and Compute the Determinant for Each unknown, Dk
• In this Example Find Dx , Dy, Dz as

50 4 22

3 6 2

33 9 12

−

−

− − − Dx =

8 4 22 50

3 6 2 3

21 9 12 33 − − + =

− + − =

− − = − x y z

x y z

x y z

8 50 22

3 3 2

21 33 12

−

− −

− − Dy =

8 4 50

3 6 3

21 9 33

− −

−

− Dz =

Cramer’s Method – Illustrated-

• Since –^ However, We can ANTICIPATE Problems if |D (^) c | << than the SMALLEST Coefficient  Completing the Example

c

k D

D

k =

 Can ID “Condition” by Calculating Dc

• SINGULAR Systems → Dc = 0
• ILL-CONDITIONED Systems → Dc = “Small” - Small is technically relative to the D (^) k

8 4 22 50

3 6 2 3

21 9 12 33

− − + =

− + − =

− − = −

x y z

x y z

x y z

Cramer’s Method – Illustrated-

• Calc the Determinants
• First Recall The SIGN pattern for Determinants

Dc = 21 ( ) (^1) −^64 − 222 − 9 ( − 1 ) (^) −− 83 − 222 − 12 ( ) (^1) −− 83 −^64

 Find Dc

1146

2604 738 720

= − −

c

c D

D

[ ( ) (( ) ( ))] [( ) (( ) ( ))] 12 [( 3 ) ( 4 ) (( 8 ) 6 )]

9 3 * 22 8 * 2

21 6 * 22 2 * 4

= − − − − −

= + − − − −

Dc = − − −

 Dc is LARGE → WELL Conditioned System

Cramer’s Method – Illustrated-

• Solve using MATLAB’s det Function >> Dz = det([EqnSys(:,1:2),EqnSys(:,4)]) Dz = 3438

>> x = Dx/Dc x = 1

>> y = Dy/Dc y = 2

>> z = Dz/Dc z = 3

All Row Elements of Cols 1-2, 4

Cramer vs Homogenous: Ax =^ b =^0

 In general, for a set of

HOMOGENEOUS linear algebraic

equations that contains the same

number of equations as unknowns

• a nonzero solution exists only if the set is singular; that is, if Cramer’s determinant is zero
• furthermore, the solution is not unique.
• If Cramer’s determinant is not zero, the homogeneous set has a zero solution; that is, all the unknowns are zero

UnderDetermined Systems

• An UNDERdetermined system does not

contain enough information to solve for ALL

of the unknown variables

• Usually because it has fewer equations than unknowns.
• In this case an INFINITE number of

solutions can exist, with one or more of the

unknowns dependent on the remaining

unknowns.

• For such systems the Matrix-Inverse and Cramer’s methods will NOT work

UnderDetermined Example-

• A simple UnderDetermined

system is the equation x +^3 y =^6

 All we can do is solve for one of the

unknowns in terms of the other; for

example, x = 6 – 3y OR y = −x/3 + 2

• An INFINITE number of (x,y) solutions satisfy this equation

More UnderDetermined Systems

• An infinite number of solutions might exist

EVEN when the number of equations EQUALS

the number of knowns

• Predict by Cramer as: (^) det ( A ) = Dc = 0

 For such systems the Matrix Inverse

method and Cramer’s method will also

NOT work

• MATLAB’s left-division method generates an error message warning us that the matrix A is singular

Minimum Norm Solution

• When det( A ) = 0, We can use the

PSEUDOINVERSE method to find ONE

Solution, x , such that the Euclidean (or

Pythagorean) Length of x is MINIMIZED

• In MATLAB:

( ) 2 2 3

2 2

2 x = min x 1 + x + x + xn

 MATLAB will return the MINIMUM

NORM SOLUTION →

x = pinv(A)*b

An INconsistent Example

• Consider Ax = b (^)  Since The Ranks

are Unequal → this system of equations is NOT solvable  Graphically

 

  

  = 

  

  

  

 5

4 2 4

1 2 2

1 x

x

 ERO: Multiply the 1 st row by −2 and add to the 2 nd^ row

 

  

 0 0

1 2 Rank[ A ]=

^ Rank[ Ab ]= 

  

 − 3

4 0

2 0

1

Existence and Uniqueness

• Recall Rank for m-Eqns & n-Unknwns

rank [ A ] = rank[ Ab ] ( ) 1

 Now Let r = rank[ A ]

• If condition (1) is satisfied and if r = n , then the solution is unique
• If condition (1) is satisfied but r < n , an infinite number of solutions exists and - r unknown variables can be expressed as linear combinations of the other n−r unknown variables, whose values are ARBITRARY