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(b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that 2₁ > 2₂ > 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 A1, A2 and 13 respectively.

(b) Show that M is diagonalisable over R and write down its three eigenvalues, ordered in such a way that 2₁ > 2₂ > 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 A1, A2 and 13 respectively.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.3: Modeling Data With Power Functions
Problem 6E: Urban Travel Times Population of cities and driving times are related, as shown in the accompanying...
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Please do questions b and c

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.
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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