Chapter 9

Let \(\mathbf{x}=[1,0,-1]^{\prime}\) and \(\mathbf{y}=[-2,1,0]^{\prime}\). (a) Find \(\mathbf{x}+\mathbf{y}\). (b) Find \(2 \mathbf{x}\). (c) Find \(-3 \mathbf{y}\).

Assume that a population is divided into three age classes and that \(20 \%\) of the females age 0 and \(70 \%\) of the females age present at time \(t\) survive to time \(t+1\). Assume further that females age 1 have an average of \(2.4\) female offspring and females age 2 have an average of \(1.3\) 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 number of females in each age class at time \(2 .\)

Let $$ A=\left[\begin{array}{rr} 2 & 2 \\ -1 & 4 \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], \quad \text { and } \quad \mathbf{y}=\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right] $$ (a) Show by direct calculation that \(A(\mathbf{x}+\mathbf{y})=A \mathbf{x}+A \mathbf{y}\). (b) Show by direct calculation that \(A(\lambda \mathbf{x})=\lambda(A \mathbf{x})\).

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Find \(A-B+2 C\)

I n ~ P r o b l e m s ~ \(1-4\), solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{array}{l} x-y=1 \\ x-2 y=-2 \end{array} $$

Let \(\mathbf{x}=[-4,3,2]^{\prime}\) and \(\mathbf{y}=[0,-2,3]^{\prime}\). (a) Find \(\mathbf{x}-\mathbf{y}\). (b) Find \(2 \mathbf{x}+3 \mathbf{y}\). (c) Find \(-\mathbf{x}-2 \mathbf{y}\).

Assume that a population is divided into three age classes and that \(80 \%\) of the females age 0 and \(10 \%\) of the females age present at time \(t\) survive until time \(t+1\). Assume further that females age 1 have an average of \(1.6\) female offspring and females age 2 have an average of \(3.9\) female offspring. If, at time 0 , the population consists of 1000 females age 0,100 females age 1 , and 20 females age 2 , find the Leslie matrix and the number of females in each age class at time \(3 .\)

Let $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], \quad \text { and } \quad \mathbf{y}=\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right] $$ (a) Show by direct calculation that \(A(\mathbf{x}+\mathbf{y})=A \mathbf{x}+A \mathbf{y}\). (b) Show by direct calculation that \(A(\lambda \mathbf{x})=\lambda(A \mathbf{x})\).

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Find \(-2 A+3 B\).

I n ~ P r o b l e m s ~ \(1-4\), solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{array}{r} 2 x+y=6 \\ x-4 y=-4 \end{array} $$

Let \(A=(2,3)\) and \(B=(1,1)\). Find the vector representation of \(\overrightarrow{A B}\).

A population is divided into four age classes. \(70 \%\) of the females age \(0,50 \%\) of the females age 1, and \(10 \%\) of the females age 2 present at time \(t\) survive until time \(t+1\). Assume 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 number of females in each age class at time \(3 .\)

Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}\) -axis (measured counterclockwise). $$\mathbf{x}=\left[\begin{array}{l}2 \\ 2\end{array}\right]$$

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Determine \(D\) so that \(A+B=2 A-B+D\).

I n ~ P r o b l e m s ~ \(1-4\), solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{aligned} x-2 y &=2 \\ y &=1+\frac{1}{2} x \end{aligned} $$

Let \(A=(-1,0)\) and \(B=(2,-3)\). Find the vector representation of \(\overrightarrow{A B}\).

A population is divided into four age classes. \(65 \%\) of the females age \(0,40 \%\) of the females age 1 , and \(30 \%\) of the females age 2 present at time \(t\) survive until time \(t+1\). Assume that females age 1 have an average of \(2.8\) female offspring, females age 2 have an average of \(7.6\) female offspring, and females age 3 have an average of \(2.4\) female offspring. If, at time 0 , the population consists of 1000 females age 0,500 females age 1,200 females age 2, and 50 females age 3, find the Leslie matrix and the number of females in each age class at time \(3 .\)

Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}\) -axis (measured counterclockwise). $$\mathbf{x}=\left[\begin{array}{r}-2 \\ 0\end{array}\right]$$

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Show that \(A+B=B+A\)

I n ~ P r o b l e m s ~ \(1-4\), solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{array}{l} 2 x+y=\frac{1}{3} \\ 6 x+3 y=1 \end{array} $$

Let \(A=(0,1,-3)\) and \(B=(-1,1,2)\). Find the vector representation of \(\overrightarrow{A B}\).

Assume the given Leslie matrix L. Determine the number of age classes in the population, the fraction of one-year-olds present at time \(t\) that survive to time \(t+1\), and the average number of female offspring of a two-year-old female. $$L=\left[\begin{array}{llll}2 & 3 & 3 & 1 \\ 0.4 & 0 & 0 & 0 \\ 0 & 0.4 & 0 & 0 \\ 0 & 0 & 0.8 & 0\end{array}\right]$$

Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}\) -axis (measured counterclockwise). $$\mathbf{x}=\left[\begin{array}{l}0 \\ 2\end{array}\right]$$

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Show that \((A+B)+C=A+(B+C)\).

Determine \(c\) such that $$ \begin{array}{l} 2 x-3 y=5 \\ 4 x-6 y=c \end{array} $$ has (a) infinitely many solutions and (b) no solutions. (c) Is it possible to choose a number for \(c\) so that the system has exactly one solution? Explain your answer.

Let \(A=(1,3,-2)\) and \(B=(0,-1,-1)\). Find the vector representation of \(\overrightarrow{A B}\).

Assume the given Leslie matrix L. Determine the number of age classes in the population, the fraction of one-year-olds present at time \(t\) that survive to time \(t+1\), and the average number of female offspring of a two-year-old female. $$L=\left[\begin{array}{lll}0 & 5 & 0 \\ 0.8 & 0 & 0 \\ 0 & 0.4 & 0\end{array}\right]$$

Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}\) -axis (measured counterclockwise). $$\mathbf{x}=\left[\begin{array}{l}-1 \\ -1\end{array}\right] \quad$$

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Find \(2 A+3 B-C\).

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Show that \(2(A+B)=2 A+2 B\).

(a) Determine the solution of $$ \begin{array}{r} -2 x+3 y=5 \\ a x-y=1 \end{array} $$ in terms of \(a\). (b) For which values of \(a\) are there no solutions, exactly one solution, and infinitely many solutions?

Find the length of \(\mathbf{x}=[2,2]^{\prime}\).

Assume the given Leslie matrix \(L\) Determine the number of age classes in the population. What fraction of two-year-olds present at time t survive until time \(t+1\). Determine the average number of female offspring of a one-year-old female. $$L=\left[\begin{array}{llll}0 & 2.5 & 4 & 1.5 \\ 0.9 & 0 & 0 & 0 \\ 0 & 0.4 & 0 & 0 \\ 0 & 0 & 0.4 & 0\end{array}\right]$$

Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}\) -axis (measured counterclockwise). $$\mathbf{x}=\left[\begin{array}{r}-\sqrt{3} \\ -1\end{array}\right]$$

Show that the solution of $$ \begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}=b_{2} \end{array} $$ where \(a_{11}, a_{12}, a_{21}, a_{22}, b_{1}\) and \(b_{2}\) are all constants, is given by $$ x_{1}=\frac{a_{22} b_{1}-a_{12} b_{2}}{a_{11} a_{22}-a_{21} a_{12}} $$ and $$ x_{2}=\frac{-a_{21} b_{1}+a_{11} b_{2}}{a_{11} a_{22}-a_{21} a_{12}} $$

Find the length of \(\mathbf{x}=[-2,7]^{\prime}\).

Assume the given Leslie matrix \(L\) Determine the number of age classes in the population. What fraction of two-year-olds present at time t survive until time \(t+1\). Determine the average number of female offspring of a one-year-old female. $$L=\left[\begin{array}{lll}0 & 4.2 & 3.7 \\ 0.7 & 0 & 0 \\ 0 & 0.1 & 0\end{array}\right]$$

Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}\) -axis (measured counterclockwise). $$\mathbf{x}=\left[\begin{array}{r}1 \\ -\sqrt{3}\end{array}\right]$$

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Find \(3 C-B+\frac{1}{2} A\).

Show that if $$ a_{11} a_{22}-a_{21} a_{12} \neq 0 $$ then the system $$ \begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=0 \\ a_{21} x_{1}+a_{22} x_{2}=0 \end{array} $$ has exactly one solution, namely, \(x_{1}=0\) and \(x_{2}=0\).

Find the length of \(\mathbf{x}=[0,1,5]^{\prime}\).

Assume that the Leslie matrix is $$ L=\left[\begin{array}{ll} 0.5 & 1.5 \\ 1 & 0 \end{array}\right] $$ 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?

Vectors are given in their polar coordinate representation (length \(\boldsymbol{r}\), and angle \(\alpha\) measured counterclockwise from the positive \(x_{1}-\) axis). Find the representation of the vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) in Cartesian coordinates. $$r=2, \alpha=60^{\circ}$$

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Determine \(D\) so that \(A+B+C+D=\mathbf{0}\).

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{r} 2 x-y=3 \\ x-3 y=7 \end{array} $$

Find the length of \(\mathbf{x}=[2,3,-1]^{\prime}\).

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?

Vectors are given in their polar coordinate representation (length \(\boldsymbol{r}\), and angle \(\alpha\) measured counterclockwise from the positive \(x_{1}-\) axis). Find the representation of the vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) in Cartesian coordinates. $$r=3, \alpha=120^{\circ}$$

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Determine \(D\) so that \(A+4 B=2(A+B)+D\).

In Problems 9-16, reduce the system of equations to upper triangular form and find all the solutions. $$ \begin{array}{l} 5 x-3 y=2 \\ 2 x+7 y=3 \end{array} $$