Gaussian%20Elimination,%20Rank%20and%20Cramer

1
Gaussian Elimination, Rank and Cramer

• We have seen how Gaussian Elimination can solve A
  x b
• But, is it always the case that there is a
  solution?
• In fact there may be many solutions.. we will
  investigate.
• This will lead to various topics
• Matrix Rank
• Homogeneous Systems
• Cramers Rule
• and finally
• Cramers Theorem
• alternative to Gaussian Elimination (for
  small matrices)

2
How Many Solutions
Consider these graphs

• Each graph has 2 lines defined by linear
  equations y - mx c
• First graph one value of x and y satisfying both
  equations, at the intersection of the lines thus
  there is one solution.
• Second graph two lines overlap - infinite
  solutions.
• Third graph two lines are parallel - there is
  no solution.
• A set of linear equations has 0, 1 or infinitely
  many solutions.
• Lets investigate systems with no solutions and
  infinite solutions.

3
e.g. 2 x y 3z 4 x y 2z
0 2 x 4 y 6 z 8
Eliminating the first column from rows 2 and
3 Row2 Row1-2Row2 2-21 1-21 3-22
4-0 0 -1 -1 4 Row3 Row1-Row3 2-2 1-4
3-6 4-8 0 -3 -3 -4
Row3 3Row2 - Row3 0 -33 33 12--4
0 0 0 16
Last row means 0 16! So, there is no solution.
4
e.g. 2 x y 3z 4 x y 2z
0 2 x 4 y 6 z -8
Eliminating the first element of rows 2 and 3
gives
Eliminating the second element of row 3 gives
The last row means 0 0 True for all values of
x, y and z. Thus there is an infinite number of
solutions to the 3 equations. In fact the
equations are said to be linearly dependent. If
there are solutions, the equations are linearly
independent.
5
Matrix Rank
The Rank of a Matrix is a property that can be
used to determine the number of solutions to a
matrix equation A x b. One definition of rank
is that it is the number of non zero rows in the
augmented matrix when it is in row echelon form.
2 rows are non zero rank 2
Here 3 rows are non zero, so rank 3. But,
matrix needed in echelon form first is there
another way?
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Rank by determinants
Rank of mn matrix is largest square submatrix
whose det ltgt 0. A submatrix of A is a matrix of
A minus some rows or columns.
Its four 33 submatrices are
a) is 2(6-8) 1(6-4) 3(4-2) -4-26
0 b) is 2(-8-0) 1(-8-0) 4(4-2)
-1688 0 c) is 2(16-0) 3(16-0) 4
(6-2) 32 48 16 0 d) is 1(-16-0)
3(-8-0) 4(6-8) -16 24 8 0
7
Rank not 3. Is it 2? Try any 22 submatrix.
Rank 2
This is the case where there is an infinite
number of solutions.
Thus Rank ( )3, but Rank(A) lt 3. Here there
are no solutions
Its rank 3
This was augmented matrix for circuit which had
one solution. Leads to Fundamental Theorem of
Linear Systems .
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Properties of Rank
Fundamental Theorem of Linear Systems
If system defined by m row matrix equation A x
b The system has solutions only if Rank (A)
Rank ( ) If Rank (A) m, there is 1
solution If Rank (A) lt m, there is an infinite
number of solutions
The Rank of A is 0 only if A is the zero
matrix. Rank (A) Rank (AT) Elementary row
operations don't affect the rank of a matrix.
Rank is a concept quite useful in control
theory. This leads to two related and useful
topics
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Homogeneous Systems If system defined by A x 0,
i.e. b 0, system is homogenous. A homogenous
system has a trivial solution, x1x2..xn0. A
non trivial solution exists if Rank(A) lt
m. Cramer's Rule For homogeneous systems If D
A ? 0, the only solution is x 0 If D 0,
the system has non trivial solutions, This is
useful, as we shall see, for eigenvalues and
eigenvectors. Cramers Theorem Solutions to a
linear system A x b , where A is an nn x1
D1/D x2 D2/D .... xn Dn/D where D is
detA , D ltgt 0, and Dk is det of matrix formed
by taking A and replacing its kth column with b.
Impracticable in large matrices as hard to find
their determinant.
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Cramers Theorem - Solving Equations
Suspended Mass
Here, D 0.96 0.6 - -0.8 0.28
0.8 Replacing first column of A with b and taking
determinant
Thus T1 240 / 0.8 300
Replacing second column of A with b and taking
determinant
Thus T2 288 / 0.8 360
These agree with earlier results. Good!
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Electronic Circuit Example
D A found earlier as 600
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Exercise
Find A such that
Use Cramers theorem to find v2 and i2 when v1
16V and i1 5A.
For given values