Chapter 9

Given are (1) a plane through \((2,0,-1)\) and perpendicular to \(\left[\begin{array}{r}-1 \\ 1 \\ 3\end{array}\right]\) and ( 2 ) a line through the points \((1,0,-2)\) and \((-1,-1,1)\). Where do the plane and the line intersect?

Use the determinant to determine whether $$ C=\left[\begin{array}{ll} 1 & 3 \\ 1 & 3 \end{array}\right] $$ is invertible. If it is invertible, compute its inverse. In either case, solve \(C X=\mathbf{0}\).

Given is a plane through \((0,-2,1)\) and perpendicular to \(\left[\begin{array}{r}-1 \\ 1 \\ -1\end{array}\right] .\) Find a line through \((5,-1,0)\) and that is parallel to the plane.

Use the determinant to determine whether $$ D=\left[\begin{array}{ll} -3 & 6 \\ -4 & 8 \end{array}\right] $$ is invertible. If it is invertible, compute its inverse. In either case, solve \(D X=\mathbf{0}\).

Find the inverse matrix to each given matrix if the inverse matrix exists. A=\left[\begin{array}{rrr} 2 & -1 & -1 \\ 2 & 1 & 1 \\ -1 & 1 & -1 \end{array}\right]

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 3 & -1 \\ 2 & -2 & 3 \\ -1 & 1 & 2 \end{array}\right] $$

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & -1 \\ 0 & -2 & 0 \\ -1 & 1 & 2 \end{array}\right] $$

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & 2 \\ -1 & -2 & 3 \\ 0 & 2 & -1 \end{array}\right] $$

Suppose that breeding occurs once a year and that a census is taken at the end of each breeding season. Assume that a population is divided into three age classes and that \(20 \%\) of the females age 0 and \(70 \%\) of the females age 1 survive until the end of the next breeding season. Assume further that females age 1 have an average of \(3.2\) female offspring and females age 2 have an average of \(1.7\) female offspring. If, at time 0, the population consists of 2000 females age 0,800 females age 1, and 200 females age 2, find the Leslie matrix and the age distribution at time 2 .

Suppose that breeding occurs once a year and that a census is taken at the end of each breeding season. Assume that a population is divided into four age classes and that \(70 \%\) of the females age \(0,50 \%\) of the females age 1, and \(10 \%\) of the females age 2 survive until the end of the next breeding season. Assume further that females age 2 have an average of \(4.6\) female offspring and females age 3 have an average of \(3.7\) female offspring. If, at time 0, the population consists of 1500 females age 0,500 females age 1,250 females age 2, and 50 females age 3, find the Leslie matrix and the age distribution at time \(2 .\)

Suppose that breeding occurs once a year and that a census is taken at the end of each breeding season. Assume that a population is divided into four age classes and that \(65 \%\) of the females age \(0,40 \%\) of the females age 1, and \(30 \%\) of the females age 2 survive until the end of the next breeding season. Assume further that females age 1 have an average of \(2.8\) female offspring, females age 2 have an average of \(7.2\) female offspring, and females age 3 have an average of 3.7 female offspring. If, at time 0, the population consists of 1500 females age 0,500 females age 1,250 females age 2, and 50 females age 3, find the Leslie matrix and the age distribution at time 3 .

$$ \begin{array}{l} \text { Let }\\\ A=\left[\begin{array}{rr} 1 & -1 / 4 \\ 1 / 2 & 1 / 4 \end{array}\right.\\\ A^{30}\left[\begin{array}{l} 1 / 2 \\ 3 / 2 \end{array}\right] \end{array} $$

Assume the given Leslie matrix \(L .\) Determine the number of age classes in the population, the fraction of oneyear-olds that survive until the end of the next breeding season, and the average number of female offspring of a two- year-old female. $$ L=\left[\begin{array}{llll} 2 & 3 & 2 & 1 \\ 0.4 & 0 & 0 & 0 \\ 0 & 0.6 & 0 & 0 \\ 0 & 0 & 0.8 & 0 \end{array}\right] $$

Suppose that $$ L=\left[\begin{array}{ll} 2 & 4 \\ 0.3 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Assume the given Leslie matrix \(L .\) Determine the number of age classes in the population, the fraction of oneyear-olds that survive until the end of the next breeding season, and the average number of female offspring of a two- year-old female. $$ L=\left[\begin{array}{lll} 0 & 5 & 0 \\ 0.8 & 0 & 0 \\ 0 & 0.3 & 0 \end{array}\right] $$

Suppose that $$ L=\left[\begin{array}{ll} 1 & 3 \\ 0.7 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Assume the given Leslie matrix \(L .\) Determine the number of age classes in the population. What fraction of two-year-olds survive until the end of the next breeding season? Determine the average number of female offspring of a one-yearold female. $$ L=\left[\begin{array}{llll} 1 & 2.5 & 3 & 1.5 \\ 0.9 & 0 & 0 & 0 \\ 0 & 0.3 & 0 & 0 \\ 0 & 0 & 0.2 & 0 \end{array}\right] $$

Suppose that $$ L=\left[\begin{array}{ll} 7 & 3 \\ 0.1 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Assume the given Leslie matrix \(L .\) Determine the number of age classes in the population. What fraction of two-year-olds survive until the end of the next breeding season? Determine the average number of female offspring of a one-yearold female. $$ L=\left[\begin{array}{lll} 0 & 4.2 & 3.7 \\ 0.7 & 0 & 0 \\ 0 & 0.1 & 0 \end{array}\right] $$

Suppose that $$ L=\left[\begin{array}{ll} 0 & 5 \\ 0.9 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Assume that the Leslie matrix is $$ L=\left[\begin{array}{ll} 1.2 & 3.2 \\ 0.8 & 0 \end{array}\right] $$ Suppose that, at time \(t=0, N_{0}(0)=100\) and \(N_{1}(0)=0 .\) Find the population vectors for \(t=0,1,2, \ldots, 10 .\) Compute the successive ratios $$ q_{0}(t)=\frac{N_{0}(t)}{N_{0}(t-1)} \quad \text { and } \quad q_{1}(t)=\frac{N_{1}(t)}{N_{1}(t-1)} $$ for \(t=1,2, \ldots, 10 .\) What value do \(q_{0}(t)\) and \(q_{1}(t)\) approach as \(t \rightarrow \infty ?\) (Take a guess.) Compute the fraction of females age 0 for \(t=0,1, \ldots, 10 .\) Can you find a stable age distribution?

Suppose that $$ L=\left[\begin{array}{ll} 0 & 5 \\ 0.09 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Assume that the Leslie matrix is $$ L=\left[\begin{array}{ll} 0.2 & 3 \\ 0.33 & 0 \end{array}\right] $$ Suppose that, at time \(t=0, N_{0}(0)=10\) and \(N_{1}(0)=5 .\) Find the population vectors for \(t=0,1,2, \ldots, 10 .\) Compute the successive ratios $$ q_{0}(t)=\frac{N_{0}(t)}{N_{0}(t-1)} \quad \text { and } \quad q_{1}(t)=\frac{N_{1}(t)}{N_{1}(t-1)} $$ for \(t=1,2, \ldots, 10 .\) What value do \(q_{0}(t)\) and \(q_{1}(t)\) approach as \(t \rightarrow \infty\) ? (Take a guess.) Compute the fraction of females age 0 for \(t=0,1, \ldots, 10 .\) Can you find a stable age distribution?