Hence state the long-term velocity of the particle.

Find the eigenvalues of $\mathit{M}$.

Find the eigenvectors of $\mathit{M}$.

Find an eigenvector corresponding to the eigenvalue of $1$. Give your answer in the form $\left(\begin{array}{c}a\\ b\end{array}\right)$, where $a, b\in \mathrm{\mathbb{Z}}$.

State geometrically what transformation the matrix $\mathbf{R}$ represents.

Find the values of $\lambda$ for which the matrix (A − $\lambda$I) is singular.

Find the eigenvalues for the matrix $\left(\begin{array}{cc}0 & 1\\ -6 & -5\end{array}\right)$.

Find the eigenvalues of the system of coupled first order equations given in part (a).

the eigenvalues.

Given that $y=\dot{x}$, show that $\dot{y}=-1.25x-3y$.

Write down a condition on the size of the initial population of $\text{Y}$ if it is to avoid its population reducing to zero.

Find the value of $t$ at which $x=0$.

Find the eigenvalues of matrix $\mathit{A}$.

Use the substitution $y=\frac{dx}{dt}$ to show that this equation can be written as

$\left(\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right)=\left(\begin{array}{cc}0 & 1\\ -6 & -5\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$.

Find the eigenvectors of matrix $\mathit{A}$.

Write down the equations of two lines of symmetry for the curve in $x$ and $y$ in part (b).

Use the result from part (b) (ii) to explain why A is non-singular.

the curve in $X$ and $Y$ in part (e) (ii), giving a reason.

Find matrix R.

Given the matrix A =  find the values of the real number $k$ for which $\text{det}\left(A-kI\right)=0$ where 

It is sunny today. Find the probability that it will be sunny in three days’ time.

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

Write down the matrix $\mathit{D}$.

the curve in $x$ and $y$ in part (b).

Find the Cartesian equation satisfied by $u$ and $v$ and state the geometric shape that this curve represents.

Show that the eigenvalues of A are real if ${\left(a-d\right)}^2+4bc⩾0$.

Determine the eigenvalues of B.

Find the eigenvalues for $M$. For each eigenvalue find the set of associated eigenvectors.

Show that the matrix equation $\left(xy\right)\mathbf{M}\left(\begin{array}{c}x\\ y\end{array}\right)=\left(6\right)$ is equivalent to the Cartesian equation $\frac{5}{2}{x}^2+xy+\frac{5}{2}{y}^2=6$.

Show that $\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{-1}{\sqrt{2}}\end{array}\right)$ and $\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$ are unit eigenvectors and that they correspond to different eigenvalues.

Use the values from part (b) (i) to express A⁴ in the form $p$A+ $q$I where $p$, $q\in \mathbb{Z}$.

Determine the corresponding eigenvectors.

Assuming that $\mathit{A}$ is non-singular, use the result in part (b)(i) to show that

${\mathit{A}}^{-1}=\frac{1}{\beta }\left(\alpha \mathit{I}-\mathit{A}\right)$.

Find the value of $m$ and of $n$.

Given that A =  and I = , find the values of $\lambda$ for which (A – $\lambda$I) is a singular matrix.

Show that .

Deduce that the eigenvalues are real if A is symmetric.

Verify that

${\mathit{A}}^2-\alpha \mathit{A}+\beta \mathit{I}=$ 0.

Find the eigenvalues and corresponding eigenvectors of the matrix $\left(\begin{array}{cc}-4& 0\\ 3& -2\end{array}\right)$.

Hence, find the Cartesian equation satisfied by $X$ and $Y$.

Verify that .

Hence show that I = $\frac{1}{5}$A (6I – A).

Hence, show that .

By considering the determinant of a relevant matrix, show that the eigenvalues, $\lambda$, of $\mathit{A}$ satisfy the equation

${\lambda }^2-\alpha \lambda +\beta =0$,

where $\alpha$ and $\beta$ are functions of $a, b, c, d$ to be determined.

Write down a transition matrix $\mathit{T}$ indicating the annual population movement between clinics.

Sketch a phase portrait for the general solution to the system of coupled differential equations for $-6\le x\le 6$, $-6\le y\le 6$.

Hence, write down the general solution of the system.

Determine, with justification, whether the equilibrium point $(0, 0)$ is stable or unstable.

Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.

Write down the matrix $\mathit{P}$.

Find a matrix $\mathit{P}$, with integer elements, such that $\mathit{T}=\mathit{P}\mathit{D}\mathbf{ }{\mathit{P}}^{-1}$, where $\mathit{D}$ is a diagonal matrix.

Hence, show that the long-term transition matrix ${\mathit{T}}^{\infty }$ is given by ${\mathit{T}}^{\infty }=\left(\begin{array}{cc}\frac{10}{17}& \frac{10}{17}\\ \frac{7}{17}& \frac{7}{17}\end{array}\right)$.

Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.

(i)   at $(4, 0)$.

(ii)  at $(-4, 0)$.

Find the population of $\text{Y}$ at this value of $t$. Give your answer to the nearest $10$ herbivores.

the eigenvectors.

Hence write down the general solution of the system of equations.

Sketch the phase portrait for this system, for $x, y>0$.

On your sketch show

• the equation of the line defined by the eigenvector in the first quadrant
• at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as t increases

Write down the matrix $\mathit{A}$.

Given that when $t=0$ the shock absorber is displaced $8 \text{cm}$ and its velocity is zero, find an expression for $x$ in terms of $t$.