Any allzero rows are at the bottom'

1
4.4.1 Generalised Row Echelon Form

• Any all-zero rows are at the bottom.
• Correct step pattern of first non-zero row
  entries.

2
4.4.1 Generalised Row Echelon Form

• Any all-zero row at the bottom
• Correct step pattern of first non-zero row
  entries

3
4.4.1 Generalised Row Echelon Form

• Any all-zero row at the bottom
• Correct step pattern of first non-zero row
  entries

4
4.4.1 Generalised Row Echelon Form

• Any all-zero row at the bottom
• Correct step pattern of first non-zero row
  entries

5
4.4.1 Generalised Row Echelon Form

• Any all-zero row at the bottom
• Correct step pattern of first non-zero row
  entries

0
0
0
1
1
3
2
2
0
0
0
0
0
0
1
0
6
4.4.1 Formal process (Handout 3)
a12
a13
a21
a23
a31
a32
7
4.4.1 Formal process (Handout 3)
a12
a13
a'23
0
a31
a32
8
4.4.1 Formal process (Handout 3)
a12
a13
a'23
0
a'32
0
9
4.4.1 Formal process
a12
a13
a'23
0
a'32
0
10
4.4.1 Formal process (Handout 3)
a12
a13
a'23
0
a'32
0
11
4.4.1 Formal process (Handout 3)
a12
a13
a'23
0
0
0
12
4.4.1 Formal process (Handout 3)
a12
a13
a'23
0
0
0
13
4.4.1 Formal process (Handout 3)
a12
a13
a'23
0
0
0
14
4.4.2 Augmented Matrix notation

• We perform row operations on the matrix and the
  opposite row

• Combine both of these into one matrix called the
  augmented matrix

15
4.4.3 Row sums

• A way to check calculations

• Add up rows
• 2. Write totals on right

9
3
7

• Do a row operation e.g.

3-2x9
16
4.4.3 Row sums

• A way to check calculations

• Add up rows
• 2. Write totals on right

• Do a row operation e.g.

9
-15
7

• Check row sums e.g. 0 (-1) 3 11 -15

17
4.5 Examples

• Matrix-vector system
• Write in augmented form

18
4.5 Examples EXAMPLE 1

• Matrix-vector system
• Write in augmented form
• Write on row sums

• Use the top left entry to create zeros below it

• First get a zero in the second row

15
36 4x15
6
19
4.5 Examples EXAMPLE 1

• Matrix-vector system
• Write in augmented form
• Write on row sums

• Use the top left entry to create zeros below it

• First get a zero in the second row

20
4.5 Examples EXAMPLE 1

• Matrix-vector system
• Write in augmented form
• Write on row sums

• Use the top left entry to create zeros below it

• First get a zero in the second row

0
0
-6
-18
21
4.5 Examples EXAMPLE 1

• Check row sums before continuing...

1 2 3 9 15 ... OK!
0 0 - 6 18 -24 ... OK!
3 1 - 2 4 6 ... OK!
22
4.5 Examples EXAMPLE 1

• Now get a zero in the third row

• Want upper triangular form so swap rows 2 and 3

23
4.5 Examples EXAMPLE 1
-11
0
-5
-23
-39

• Now solve by backwards substitution

-5y -33 -23
r1 x 2y 3z 9

• Hence x 4, y -2, z 3 is the unique
  solution.

24
4.5 Examples

• Matrix-vector system
• Write in augmented form

25
4.5 Examples EXAMPLE 2

• Matrix-vector system
• Write in augmented form
• Write on row sums

• Use the top left entry to create zeros below it

• First get a zero in the third row

26
4.5 Examples EXAMPLE 2

• Use second row to get a zero in the third row

27
4.5 Examples EXAMPLE 2

• Solve by backwards substitution

r3 2z -8
r2 y - z 6
r1 x - y 1
28
4.6 Determinants
Question During the elimination process, what
has changed about the determinant of the matrix?

• Swapping rows multiplies the determinant by (-1)

• Adding or subtracting multiples of rows does not
  change the determinant

29
4.6 Determinants

• In EXAMPLE 1 we used one swap operation to get
  from

B
A

• Calculating the determinant

A (-1) B
(-1)x(1)(-5)(-6) -30

• Non-zero, so we got a unique solution

30
4.6 Determinants

• In EXAMPLE 2 we used no swaps to get from

B
A

• Calculating the determinant

(1)(1)(2) 2
A B

• Non-zero, so got a unique solution

31
4.6 Non-Standard Gaussian Elimination

• In standard Gaussian Elimination the following
  operation were allowed

• Swap two rows
• Add or Subtract a multiple of a row from another
  row.

• In Non-Standard Gaussian Elimination we are also
  allowed to do the following

• Multiply a row by a constant. E.g.

32
4.6 Non-Standard Gaussian Elimination

• Quick Example In Standard G.E.

• This is a bit messy with the fractions. However,
  in Non-Standard G.E.

• However, in doing this we have multiplied the
  determinant by 3.

33
4.7 Backwards substitution more general case

• Two cases after elimination process

• All diagonal entries non-zero, then determinant
  is non-zero. Hence, get answer by backwards
  substitution.
• Is a zero on the diagonal, then determinant is
  zero. Either get infinite solutions or no
  solutions.

34
4.7.1 Case of No Solutions

• Suppose we followed the elimination process and
  got to

11
-3
9

• Zeros on the diagonal, so determinant is zero.

• ROW 3 gives the equation

• This is impossible. Hence there are no solutions.

35
4.7.1 Case of Infinite solutions

• Suppose instead that

• ROW 3 now OK 0x 0y 0z 0

• Have two equations in three unknowns

• Get infinitely many solutions

36
4.7.1 Case of Infinite solutions

• Three steps

• In the final (echelon-form) of the matrix, circle
  the first non-zero entry in each row
• Find the columns that have no circles in. Each
  column corresponds to a variable.
• Assign a new name to each of the chosen
  variables, then use back substitution on the
  non-zero rows.

37
4.7.1 Case of Infinite solutions

• Circle first non-zero row entries
• Find column with no circles in

Column 2 corresponds to the y variable

• 3. Assign a name to y let y a

38
4.7.1 Case of Infinite solutions

• Solve by back substitution

r3 Tells us nothing
r1 3x y 2z 11