In this question, the real numbers in your answers should be given with three significant digits of accuracy after the decimal point. 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 λ1>λ2>λ3. (c) Write down eigenvectors v1,v2,v3 attached toλ1, λ2 and λ3 respectively.
In this question, the real numbers in your answers should be given with three significant digits of accuracy after the decimal point.
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:
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Among those who are negative, 95% remain so, 4% become positive, and 1% need to be hospitalised.
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Among those who are positive, 75% recover and become negative, 20% stay positive, and 5% need to be hospitalised.
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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 λ1>λ2>λ3.
(c) Write down eigenvectors v1,v2,v3 attached toλ1, λ2 and λ3 respectively.
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