Eigenvalues & Eigenvectors (DP IB Applications & Interpretation (AI): HL): Exam Questions

Consider the $2\times 2$ matrix A defined by

$\mathit{A}=\left(\begin{array}{cc}0.35& 0.15\\ 0.65& 0.85\end{array}\right)$ 

(i) Find the characteristic polynomial of A.

(ii) Find the eigenvalues of A.

Let ${\lambda }_1$and ${\lambda }_2$ be the eigenvalues found in part (a)(ii), and let ${\mathit{x}}_1$ and ${\mathit{x}}_2$ be the eigenvectors of A corresponding to ${\lambda }_1$and ${\lambda }_2$respectively.

Find eigenvectors ${\mathit{x}}_1$ and ${\mathit{x}}_2$.

Explain with justification whether the answers found in part (b) are unique.

Consider the matrix A defined as

$\mathit{A}=\left(\begin{array}{cc}-1& 0.75\\ 4& -1.5\end{array}\right)$ 

Find the eigenvalues and corresponding eigenvectors of matrix A.

Now consider the matrix $k\mathit{A}$defined as

 $k\mathit{A}=\left(\begin{array}{cc}-k& 0.75k\\ 4k& -1.5k\end{array}\right)$ 

where $k\ne 0$ is a real constant. 

Show that the eigenvectors found in part (a) are also eigenvectors of matrix $k\mathit{A}$ and determine their corresponding eigenvalues.

Consider the matrix $\mathit{B}$ defined as

 $\mathit{B}=\left(\begin{array}{cc}6& -2\\ 1& 2\end{array}\right)$ 

Find the eigenvalues and corresponding eigenvectors of $\mathit{B}$.

Find the eigenvalues for each of the following matrices:

$\mathit{C}=\left(\begin{array}{cc}5& -4\\ 4.5& -1\end{array}\right)$

$\mathit{D}=\left(\begin{array}{cc}5k& -4k\\ 8.5k& -5k\end{array}\right)$

where $k\ne 0$ is a real constant.

Show that $\left(\begin{array}{c}2+2\mathrm{i}\\ 3\end{array}\right)$    is an eigenvector of matrix $\mathit{C}$, and find the other eigenvector.

Consider the matrix $\mathit{M}$defined as

 $\mathit{M}=\left(\begin{array}{cc}-3& k\\ 2& 6\end{array}\right)$ 

where $k\in \mathrm{ℝ}$ is a constant. 

Given that -2 is an eigenvalue of $\mathit{M}$, 

find the remaining eigenvalue of $\mathit{M}$, as well as the eigenvectors that correspond to the two eigenvalues.

Hence diagonalise $\mathit{M}$ by writing it in the form $\mathit{P}\mathit{D}{\mathit{P}}^{\mathbf{-}\mathbf{1}}$ for appropriate matrices $\mathit{P}$ and $\mathit{D}$.

It is given that, for $n\times n$ matrices $\mathit{A},\mathit{B}$ and $\mathit{C}$,

 $\mathit{A}=\mathit{B}\mathit{C}{\mathit{B}}^{-1}$

Use the properties of matrices and matrix inverses to explain why  ${\mathit{A}}^n=\mathit{B}{\mathit{C}}^n{\mathit{B}}^{-1}$.

Consider the matrix  $\mathit{M}=\left(\begin{array}{cc}p& 1\\ -2& -3\end{array}\right)$,  where $p\in \mathrm{ℝ}$ is a constant and where it is given that $\left(\begin{array}{c}1\\ 1\end{array}\right)$ is an eigenvector of $\mathit{M}$.

By first finding the eigenvalues and the other eigenvector of $\mathit{M}$, write $\mathit{M}$ in the form $\mathit{M}\mathbf{=}\mathit{P}\mathit{D}{\mathit{P}}^{\mathbf{-}\mathbf{1}}$   for appropriate matrices $\mathit{P}$and $\mathit{D}$.

(i) Use the result of part (b) to show that

 $M^n={\left(-1\right)}^n\left(\begin{array}{cc}2\times 5^n& 4^n-5^n\\ 2\left(5^n-4^n\right)& 2\times 4^n-5^n\end{array}\right)$

(ii) Show that the expression for ${\mathit{M}}^n$ in part (c)(i) gives the expected result when $n=3.$ 

Exobiologists are studying two species of animals in a region of the distant planet Dirion.  In the researchers’ models the population of Reddors (a prey species) is indicated by r, while the population of Sklyveths (a predator species that preys on Reddors) is indicated by s. 

If the respective populations at a particular point in time are $r_n$ and $s_n$, then the researchers’ data suggest that the populations one year later may be modelled by the following system of coupled equations:

$r_{n+1}=1.3r_n-0.25s_n$

$s_{n+1}=0.07r_n+0.9s_n$

Represent the system of equations in the matrix form ${\mathit{x}}_{n+1}=\mathit{M}{\mathit{x}}_n$.

By first finding the eigenvalues and eigenvectors of $\mathit{M}$, write $\mathit{M}$ in the form $\mathit{M}=\mathit{P}\mathit{D}{\mathit{P}}^{\mathbf{-}\mathbf{1}}$ for appropriate matrices $\mathit{P}$ and $\mathit{D}$.

At the start of the study there were 2100 Reddors and 2850 Sklyveths in the region. 

Show that the respective populations after $n$ years are predicted by the model to be $r_n=75\left(1.{25}^n\right)+2025\left(0.{95}^n\right)$ and $s_n=15\left(1.{25}^n\right)+2835\left(0.{95}^n\right)$.

(i) Determine the ratio of Reddors to Sklyveths that the model predicts will be in the region in the long term.  Be sure to justify your answer.

(ii) Determine the number of years it will take after the start of the study for the population of Reddors to exceed the population of Sklyveths.

Find the eigenvalues and corresponding eigenvectors for each of the following matrices:

$A=\left(\begin{array}{cc}2& -\frac{7}{3}\\ \frac{13}{6}& -\frac{5}{2}\end{array}\right)$

$B=\left(\begin{array}{cc}0.1& 0.5\\ -0.02& 0.3\end{array}\right)$

Find the eigenvalues and corresponding eigenvectors for each of the following matrices:

$C=\left(\begin{array}{cc}-3& 17\\ -2& 3\end{array}\right)$

$D=\left(\begin{array}{cc}1.5& -2.5\\ 4.5& 2.5\end{array}\right)$

Let $M=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ be a $2\times 2$ matrix with real-valued elements $a,b,c$ and $d$.  Let ${\lambda }_1$ and ${\lambda }_2$ be the eigenvalues of matrix $M$.

In the case where ${\lambda }_1\ne {\lambda }_2$ ,  show that 

(i) $a+d={\lambda }_1+{\lambda }_2$

(ii) $\det M={\lambda }_1{\lambda }_2$

In the case where ${\lambda }_1={\lambda }_2$,  show that ${\left(a-d\right)}^2+4bc=0$.

Hence show that the results of part (a) are also true when matrix M has a single repeated eigenvalue.

Let  $M=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$  be a  matrix with real-valued elements $a,b,c$ and $d$ which are such that  $a+c=1$  and $b+d=1$.

Show that the eigenvalues of $M$ are 1 and $\left(a+d\right)-1$. 

In the case where $M\ne I$ ,  find the eigenvectors of $M$ corresponding to the eigenvalues found in part (a).  Give your answers, where appropriate, in terms of a and d only. 

In the case where $M=I$ ,  describe briefly the eigenvalues and eigenvectors of $M$.

Consider the matrix $M$ defined as

 $M=\left(\begin{array}{cc}\frac{23}{14}& \frac{5}{7}\\ \frac{15}{14}& k\end{array}\right)$ 

where $k\in \mathrm{ℝ}$ is a constant.  It is given that  $-\frac{1}{2}$ is an eigenvalue of $M$. 

By first finding the value of $k$, diagonalise $M$ by writing it in the form $PD{P}^{-1}$ for appropriate matrices  $P$ and $D$.

Consider the matrix $M=\left(\begin{array}{cc}p& 1\\ 2& q\end{array}\right)$,  where $p,q\in \mathrm{ℝ}$ are constants.

It is given that -6 is an eigenvalue of $M$, and also that $\left(\begin{array}{c}1\\ 2\end{array}\right)$ is an eigenvector of $M$ which does not correspond to the eigenvalue -6.

By first finding the values of $p$ and $q$, write $M$ in the form $M=PD{P}^{-1}$ for appropriate matrices $P$ and $D$. 

Hence show that

$M^n=\frac{{\left(-1\right)}^n}{3}\left(\begin{array}{cc}{3}^n+2\times 6^n& 3^n-6^n\\ 2\left(3^n-6^n\right)& 2\times 3^n+6^n\end{array}\right)$

Two towns, Avaricia and Covetton, are located on opposite sides of a national park. The two towns are heavily dependent on tourism, and they compete with one another both for the business of tourists coming to the park, and for residents to work in the tourism industry. 

Government officials studying the two towns indicate the population of Avaricia by a, and the population of Covetton by c.  If the respective populations at a particular point in time are $a_n$and $c_n$, then data suggest that the populations one year later may be modelled by the following system of coupled equations:

$a_{n+1}=1.025a_n-0.075c_n$

$c_{n+1}=-0.025a_n+0.975c_n$

Let $a_0$ and $c_0$ indicate the respective populations of the two towns at the start of the study. 

Use a matrix method to show that the respective populations after $n$ years are predicted by the model to be 

$a_n=0.75\left(a_0-c_0\right)\left(1.{05}^n\right)+\left(0.25a_0+0.75c_0\right)\left(0.{95}^n\right)$

$c_n=0.25\left(c_0-a_0\right)\left(1.{05}^n\right)+\left(0.25a_0+0.75c_0\right)\left(0.{95}^n\right)$

Describe what the model predicts in the long term for the populations of the two towns, for each of the following situations:

i) $a_0=c_0$

ii) $a_0>c_0$

iii) $a_0<c_0$