The public health authorities of a small town have divided the population into three categories: covid- negative, covid-positive, and hospitalised. After performing regular, extensive tests, they have observed that in each successive week: • Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised. • Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to be hospitalised. • Among those who are hospitalised, 60% recover and become negative, 30% are released from hospital but remain positive, and 10% remain hospitalised. (a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h represent the number of people in the population who are negative, positive, and hospitalised respectively, write down a matrix M for which [n', p', h'] = M[n; p; h], where [n', p', h'] represents the column vector of negative, positive, and hospitalised members of the population in the following week. (b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that A1 > A₂ > 23. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen command in Pari-GP.) (c) Write down eigenvectors V₁, V2, V3 attached to λ1, 2 and 23 respectively.
The public health authorities of a small town have divided the population into three categories: covid- negative, covid-positive, and hospitalised. After performing regular, extensive tests, they have observed that in each successive week: • Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised. • Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to be hospitalised. • Among those who are hospitalised, 60% recover and become negative, 30% are released from hospital but remain positive, and 10% remain hospitalised. (a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h represent the number of people in the population who are negative, positive, and hospitalised respectively, write down a matrix M for which [n', p', h'] = M[n; p; h], where [n', p', h'] represents the column vector of negative, positive, and hospitalised members of the population in the following week. (b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that A1 > A₂ > 23. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen command in Pari-GP.) (c) Write down eigenvectors V₁, V2, V3 attached to λ1, 2 and 23 respectively.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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![The public health authorities of a small town have divided the population into three categories: covid-
negative, covid-positive, and hospitalised. After performing regular, extensive tests, they have observed
that in each successive week:
• Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised.
• Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to
be hospitalised.
• Among those who are hospitalised, 60% recover and become negative, 30% are released from
hospital but remain positive, and 10% remain hospitalised.
(a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h
represent the number of people in the population who are negative, positive, and hospitalised respectively,
write down a matrix M for which [n', p', h'] = M[n; p; h], where [n', p', h'] represents the column
vector of negative, positive, and hospitalised members of the population in the following week.
(b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that
A1 > 12 > A3. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen
command in Pari-GP.)
(c) Write down eigenvectors v1, V2, V3 attached to 11, 12 and 13 respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb347f45-894f-4931-8337-0abefb8437d3%2F4fd84f33-0db7-48f6-b5e4-4c010e00a4cd%2Ffoq479_processed.png&w=3840&q=75)
Transcribed Image Text:The public health authorities of a small town have divided the population into three categories: covid-
negative, covid-positive, and hospitalised. After performing regular, extensive tests, they have observed
that in each successive week:
• Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised.
• Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to
be hospitalised.
• Among those who are hospitalised, 60% recover and become negative, 30% are released from
hospital but remain positive, and 10% remain hospitalised.
(a) After representing the population in a given week as a column vector v = [n; p; h], where n, p, and h
represent the number of people in the population who are negative, positive, and hospitalised respectively,
write down a matrix M for which [n', p', h'] = M[n; p; h], where [n', p', h'] represents the column
vector of negative, positive, and hospitalised members of the population in the following week.
(b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that
A1 > 12 > A3. (You may use the computer to do this calculation if you prefer, e.g., by using the mateigen
command in Pari-GP.)
(c) Write down eigenvectors v1, V2, V3 attached to 11, 12 and 13 respectively.
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