SCHOOL OF MATHEMATICS
TRINITY COLLEGE DUBLIN
Markov Matrices 1S2
JF Natural Sciences
Square matrices with non-negative entries whose columns sum to one are called Markov matrices.
In preparation for the first tutorial next term, attempt and reflect on the following questions.
1. Within a particular geographical region a census is taken at equal times . . . , −2T , −T , 0,
T , 2T , . . . , . The following population movements are observed in each time period T
(a) 80% of cities, 20% of towns, 10% of rural areas, move to the cities
(b) 10% of cities, 70% of towns, 30% of rural areas, move to the towns
(c) 10% of cities, 10% of towns, 60% of rural areas, move to the rurals.
If at a particular time the number of people in cities is 20,000 and the population level in
towns is 40,000 while the numbers living in rural areas is 40,000, calculate the population
levels to be expected in cities, towns, and rural areas after a time period T has elapsed.
2. The following matrix has elements formed from the percentages given above. Its rows and
columns are labelled in the order cities, towns, and rural areas. Show that the product of
the matrix M with the column vector of population values X, in units of a thousand for
convenience, reproduces in the correct order the population levels that have been evaluated.
M =
1
10





8
2
1
1
7
3
1
1
6



,


20
X = 


40

40



.

3. Find the product of the matrix M with the inverse matrix, M −1 , calculated below, namely
M −1 =
1
30





39
−11
−1
−3
47
−23
−6
−6
54



.

Is the product of the inverse matrix M −1 and the matrix M the unit three by three matrix?
4. Calculate the product of the matrix M −1 with the column vector of population values X to
provide the population levels of the previous time period.
5. Verify that the product of the matrix M with the column vector of population values of the
previous time period, M −1 X, correctly reproduces the above column vector of population
values X.
6. Would you expect that the product of the matrix M −1 with column vector of population
values M X calculated at the beginning should be the original column vector X ?
Enjoy the end of the tercententary year of the naming of Archimedes’ ratio, π, by W. Jones in 1706
and the beginning of the sesquiquatercentenary of the appearance in The Whetstone of Witte of
the equality symbol “=” introduced for the first time in a book written by Robert Recorde in 1557.
Dr. Buttimore
www.maths.tcd.ie/~nhb/1S2.php
The Inverse of a Square Matrix from using a Reduced Row Echelon Form
The inverse of the three by three matrix M is found by obtaining the reduced row echelon form of
the given square matrix augmented by a unit matrix, seeking zeros above and below the diagonal:




0.2
0.1
1
0
0
0.1
0.7
0.3
0
1
0
0.1
0.1
0.6
0
0
1
0
10

1




1
6
0



3
0
10
0
8
2
1
10
0
0
6
0
0
10



0
6
−3
0
10
−10
8
2
1
10
0
0

1



1
6
0
Use 1 in first column to reduce rows

7
1
Swap rows 1 and 3 to put a 1 at top



1
1
10 × each row to clear decimal fractions

0.8
0





10
0
6
−3
0
10
−10
0
−6
−47
10
0
−80



−1 × Row 1
−1
−1
−6
0
0
−10
Row 2
1
7
3
0
10
0
New Row 2
0
6
−3
0
10
−10
−8 × Row 1
−8
−8
−48
0
0
−80
Row 3
8
2
1
10
0
0
New Row 3
0
−6
−47
10
0
−80
Row 2
0
6
−3
0
10
−10
Row 3
0
−6
−47
10
0
−80
New Row 3
0
0
−50
10
10
−90
Divide the new third row here by −10 to simplify the third row as indicated on the left in the following

1







0
0
−3
0
10
−10
0
0
5
−1
−1
9
1
1
6
0

10
6






6
0


1
0



0
30
0
−3
47
−23
0
0
5
−1
−1
9
5
5
0
6
6
−4
0
30
0
−3
47
−23
0
0
5
−1
−1
9
30
0
0
39
−11
0
30
0
−3
47
0
0
30
−6
−6
Dr. Buttimore

10







−1


−23 

54
1S2
5 × Row 2
0
30
−15
0
50
−50
3 × Row 3
0
0
15
−3
−3
27
New Row 2
0
30
0
−3
47
−23
5 × Row 1
5
5
30
0
0
50
−6 × Row 3
0
0
−30
6
6
−54
New Row 1
5
5
0
6
6
−4
6 × Row 1
30
30
0
36
36
−24
−1 × Row 2
0
−30
0
3
−47
23
New Row 1
30
0
0
39
−11
−1
6 × Row 3
0
0
30
−6
−6
54
New Row 3
0
0
30
−6
−6
54
All Rows/30
End
nhb @ maths.tcd.ie