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
Cramer’s Rule
and finally
Cramer’s Theorem
– alternative to Gaussian Elimination (for small matrices)
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EG1C2 Engineering Maths: Matrix Algebra 6

2. EG1C2 Engineering Maths: Matrix Algebra 6
How Many Solutions
Consider these graphs
Each graph has 2 lines defined by linear equations: y - m*x = 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.
Let’s investigate systems with no solutions and infinite solutions.
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EG1C2 Engineering Maths: Matrix Algebra 6

3. EG1C2 Engineering Maths: Matrix Algebra 6
e.g. 2 x + y + 3z = 4; x + y + 2z = 0; 2 x y z = 8
Eliminating the first column from rows 2 and 3:
Row2 := Row1-2*Row2 [2-2*1 1-2*1 3-2*2 4-0] = [ ]
Row3 := Row1-Row3 [ ] = [ ]
Row3 := 3*Row2 - Row3 [0 -3—3 –3— ] = [ ]
Last row means 0 = 16! So, there is no solution.
p3 RJM 15/10/02
EG1C2 Engineering Maths: Matrix Algebra 6

4. Eliminating the first element of rows 2 and 3 gives
e.g. 2 x + y + 3z = 4; x + y + 2z = 0; 2 x y 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.
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EG1C2 Engineering Maths: Matrix Algebra 6

5. EG1C2 Engineering Maths: Matrix Algebra 6
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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EG1C2 Engineering Maths: Matrix Algebra 6

6. EG1C2 Engineering Maths: Matrix Algebra 6
Rank by determinants
Rank of m*n matrix is largest square submatrix whose det <> 0.
A submatrix of A is a matrix of A minus some rows or columns.
Its four 3*3 submatrices are:
| a) | is 2*(6-8) –1*(6-4) + 3*(4-2) = = 0
| b) | is 2*(-8-0) –1*(-8-0) + 4*(4-2) = = 0
| c) | is 2*(16-0) –3*(16-0) + 4* (6-2) = 32 – = 0
| d) | is 1*(-16-0) – 3*(-8-0) + 4*(6-8) = – 8 = 0
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EG1C2 Engineering Maths: Matrix Algebra 6

7. EG1C2 Engineering Maths: Matrix Algebra 6
Rank not 3. Is it 2? Try any 2*2 submatrix.
Rank = 2
This is the case where there is an infinite number of solutions.
Thus Rank ( )=3, but Rank(A) < 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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EG1C2 Engineering Maths: Matrix Algebra 6

8. 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) < m, there is an infinite number of solutions
Properties of Rank
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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EG1C2 Engineering Maths: Matrix Algebra 6

9. EG1C2 Engineering Maths: Matrix Algebra 6
Homogeneous Systems
If system defined by A x = 0, i.e. b = 0, system is homogenous.
A homogenous system has a trivial solution, x1=x2=..xn=0.
A non trivial solution exists if Rank(A) < 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.
Cramer’s Theorem
Solutions to a linear system A * x = b , where A is an n*n:
x1 = D1/D x2 = D2/D xn = Dn/D
where D is det|A| , D <> 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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EG1C2 Engineering Maths: Matrix Algebra 6

10. Cramer’s Theorem - Solving Equations
Suspended Mass
Here, D = 0.96 * * 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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EG1C2 Engineering Maths: Matrix Algebra 6

11. EG1C2 Engineering Maths: Matrix Algebra 6
Electronic Circuit Example
D = |A| found earlier as –600
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EG1C2 Engineering Maths: Matrix Algebra 6

12. EG1C2 Engineering Maths: Matrix Algebra 6
Exercise
Find A such that
Use Cramer’s theorem to find v2 and i2 when v1 = 16V and i1 = 5A.
For given values:
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EG1C2 Engineering Maths: Matrix Algebra 6