The matrix \(\mathbf A\) is given by
\(\mathbf A=\begin{pmatrix}2&2&-3\\2&2&3\\-3&3&3\end{pmatrix}\).
The matrix \(\mathbf A\) has an eigenvector \(\begin{pmatrix}1\\-1\\1\end{pmatrix}\). Find the corresponding eigenvalue.
The matrix \(\mathbf A\) also has eigenvalues \(4\) and \(6\). Find corresponding eigenvectors.
Hence find a matrix \(\mathbf P\) and a diagonal matrix \(\mathbf D\) such that \(\mathbf A=\mathbf P\mathbf D\mathbf P^{-1}\).
The matrix \(\mathbf B\) is such that \(\mathbf B=\mathbf Q\mathbf A\mathbf Q^{-1}\), where \(\mathbf Q=\begin{pmatrix}4&11&5\\1&4&2\\1&2&1\end{pmatrix}\).
By using the expression \(\mathbf P\mathbf D\mathbf P^{-1}\) for \(\mathbf A\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf B\).
EITHER
It is given that \(1\) and \(4\) are eigenvalues of the matrix \(\mathbf A\), where
\(\mathbf A=\begin{pmatrix}1&-3&-3\\-8&6&-3\\8&-2&7\end{pmatrix}.\)
Find eigenvectors corresponding to each of these eigenvalues.
Given further that \(\begin{pmatrix}0\\1\\-1\end{pmatrix}\) is an eigenvector of \(\mathbf A\), find the corresponding eigenvalue.
Write down matrices \(\mathbf P\) and \(\mathbf D\) such that \(\mathbf P^{-1}\mathbf A\mathbf P=\mathbf D\), where \(\mathbf D\) is a diagonal matrix, and find \(\mathbf P^{-1}\).
Write down a matrix \(\mathbf C\) such that \(\mathbf C^2=\mathbf D\), and deduce a matrix \(\mathbf B\) such that \(\mathbf B^2=\mathbf A\).
Write down the eigenvalues of the matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & 1 \end{array}\right),\)
and find corresponding eigenvectors.
Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{P}^{-1} \mathbf{A P}=\mathbf{D}\), and hence find the matrix \(\mathbf{A}^{n}\), where \(n\) is a positive integer.
[Question 11 is printed on the next page.]
The matrix \(\mathbf{A}\) is defined by
\(\mathbf{A}=\left(\begin{array}{rrr}
1 & 5 & 1 \\
1 & -2 & -2 \\
2 & 3 & \theta
\end{array}\right) .\)
(i) (a) Find the rank of \(\mathbf{A}\) when \(\theta \neq-1\).
(b) Find the rank of \(\mathbf{A}\) when \(\theta=-1\).
Consider the system of equations
\(\begin{aligned}
x+5 y+z & =-1 \\
x-2 y-2 z & =0 \\
2 x+3 y+\theta z & =\theta
\end{aligned}\)
(ii) Solve the system of equations when \(\theta \neq-1\).
(iii) Find the general solution when \(\theta=-1\).
(iv) Show that if \(\theta=-1\) and \(\phi \neq-1\) then \(\mathbf{A x}=\left(\begin{array}{r}-1 \\ 0 \\ \phi\end{array}\right)\) has no solution.
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{ccc} a & -6 a & 2 a+2 \\ 0 & 1-a & 0 \\ 0 & 2-a & -1 \end{array}\right)\)
where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).
(a) Show that the equation \(\mathbf{A}\left(\begin{array}{c}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)\) has a unique solution and interpret this situation geometrically.
(b) Show that the eigenvalues of \(\mathbf{A}\) are \(a, 1-a\) and -1 .
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{4}=\mathbf{P D P}^{-1}\).
(d) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{4}\) in terms of \(\mathbf{A}\) and \(a\).
(a) Show that the system of equations
\(\begin{array}{r} x+2 y+3 z=1 \\ 4 x+5 y+6 z=1 \\ 7 x+8 y+9 z=1 \end{array}\)
does not have a unique solution.
(b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
Find the set of values of \(k\) for which the system of equations
\(\begin{array}{r} x+5 y+6 z=1 \\ k x+2 y+2 z=2 \\ -3 x+4 y+8 z=3 \end{array}\)
has a unique solution and interpret this situation geometrically.
Show that the system of equations
\(\begin{aligned} 14 x-4 y+6 z & =5 \\ x+y+k z & =3 \\ -21 x+6 y-9 z & =14 \end{aligned}\)
where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.
(a) Find the value of \(a\) for which the system of equations
\(\begin{array}{c} 3 x+a y=0 \\ 5 x-y=0 \\ x+3 y+2 z=0 \end{array}\)
does not have a unique solution.
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 5 & -1 & 0 \\ 1 & 3 & 2 \end{array}\right)\)
(b) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{2}=\mathbf{P D P}^{-1}\).
(c) Use the characteristic equation of \(\mathbf{A}\) to show that
\((\mathbf{A}+6 \mathbf{I})^{2}=\mathbf{A}^{4}(\mathbf{A}+b \mathbf{I})^{2}\)
where \(b\) is an integer to be determined.
(a) Show that the system of equations
\(\begin{array}{l} x-y+2 z=4 \\ x-y-3 z=a \\ x-y+7 z=13 \end{array}\)
where \(a\) is a constant, does not have a unique solution.
(b) Given that \(a=-5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
(c) Given instead that \(a \neq-5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
(a) Find the set of values of \(k\) for which the system of equations
\(\begin{aligned} x+2 y+3 z & =1 \\ k x+4 y+6 z & =0 \\ 7 x+8 y+9 z & =3 \end{aligned}\)
has a unique solution.
(a) Given that \(a\) is an integer, show that the system of equations
\(\begin{aligned} a x+3 y+z & =14 \\ 2 x+y+3 z & =0 \\ -x+2 y-5 z & =17 \end{aligned}\)
has a unique solution and interpret this situation geometrically.
(b) Find the value of \(a\) for which \(x=1, y=4, z=-2\) is the solution to the system of equations in part (a).
The matrix \(A\), given by \(A=\begin{pmatrix}1&-1&0&2\\3&-1&4&0\\5&-8&-6&19\\-2&3&2&-7\end{pmatrix}\), represents a transformation from \(\mathbb{R}^4\) to \(\mathbb{R}^4\).
(i) Find the rank of \(A\) and show that \(\left\{\begin{pmatrix}2\\2\\-1\\0\end{pmatrix},\begin{pmatrix}1\\3\\0\\1\end{pmatrix}\right\}\) is a basis for the null space of the transformation.
(ii) Show that if \(A\mathbf{x}=p\begin{pmatrix}1\\3\\5\\-2\end{pmatrix}+q\begin{pmatrix}-1\\-1\\-8\\3\end{pmatrix}\), where \(p\) and \(q\) are given real numbers, then \(\mathbf{x}=\begin{pmatrix}p+2\lambda+\mu\\q+2\lambda+3\mu\\-\lambda\\\mu\end{pmatrix}\), where \(\lambda\) and \(\mu\) are real numbers.
(iii) Find the values of \(p\) and \(q\) such that \(p\begin{pmatrix}1\\3\\5\\-2\end{pmatrix}+q\begin{pmatrix}-1\\-1\\-8\\3\end{pmatrix}=\begin{pmatrix}3\\7\\18\\-7\end{pmatrix}\).
(iv) Find the solution of \(A\mathbf{x}=\begin{pmatrix}3\\7\\18\\-7\end{pmatrix}\) of the form \(\mathbf{x}=\begin{pmatrix}4\\9\\m\\n\end{pmatrix}\), where \(m\) and \(n\) are positive integers to be found.
Find the value of the constant \(k\) for which the system of equations
\(\begin{aligned} 2 x-3 y+4 z & =1 \\ 3 x-y & =2 \\ x+2 y+k z & =1 \end{aligned}\)
does not have a unique solution.
For this value of \(k\), solve the system of equations.
Find the two values of the constant \(k\) for which the equations
\(\begin{aligned}kx+y+z&=2\\x+ky+z&=-1\\x+y+kz&=-1\end{aligned}\)
have no unique solution.
Show that, for one of these values of \(k\), the equations have no solution, and solve the equations for the other value of \(k\).
OR
The linear transformation \(T:\mathbb R^4\to\mathbb R^4\) is represented by the matrix
\(\mathbf M=\begin{pmatrix}1&-2&3&-4\\2&-4&7&-9\\4&-8&14&-18\\5&-10&17&-22\end{pmatrix}.\)
Find the rank of \(\mathbf M\).
Obtain a basis for the null space \(K\) of \(T\).
Evaluate
\(\mathbf M\begin{pmatrix}1\\-2\\2\\-1\end{pmatrix},\)
and hence show that any solution of
\(\mathbf M\mathbf x=\begin{pmatrix}15\\33\\66\\81\end{pmatrix}\)
has the form
\(\begin{pmatrix}1\\-2\\2\\-1\end{pmatrix}+\lambda\mathbf e_1+\mu\mathbf e_2,\)
where \(\lambda\) and \(\mu\) are scalars and \(\{\mathbf e_1,\mathbf e_2\}\) is a basis for \(K\). Hence obtain a solution \(\mathbf x'\) for which the sum of the components is \(6\) and the sum of the squares of the components is \(26\).
The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{rrrr} 1 & -1 & -2 & 3 \\ 5 & -3 & -4 & 25 \\ 6 & -4 & -6 & 28 \\ 7 & -5 & -8 & 31 \end{array}\right) .\)
(i) Find the rank of \(\mathbf{A}\) and a basis for the null space of T .
(ii) Find the matrix product \(\mathbf{A}\left(\begin{array}{r}-1 \\ 1 \\ -1 \\ 1\end{array}\right)\) and hence find the general solution of the equation \(\mathbf{A} \mathbf{x}=\left(\begin{array}{r}3 \\ 21 \\ 24 \\ 27\end{array}\right)\).
Show that the matrix \(\left(\begin{array}{rrr}1 & 0 & -2 \\ 3 & -3 & -4\end{array}\right)\) has no inverse.
Solve the system of equations
\(\begin{array}{r} x+4 y+2 z=0 \\ 3 x-2 z=4 \\ 3 x-3 y-4 z=5 \end{array}\)
1 A curve \(C\) has equation \(\cos y=x\), for \(-\pi<x<\pi\).
(i) Use implicit differentiation to show that \(\dfrac{\mathrm d^2y}{\mathrm dx^2}=-\cot y\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2\).
(ii) Hence find the exact value of \(\dfrac{\mathrm d^2y}{\mathrm dx^2}\) at the point \(\left(\dfrac12,\dfrac{\pi}{3}\right)\) on \(C\).
Answer only one of the following two alternatives.
EITHER
It is given that \(w=\cos y\) and
\(\tan y\,\dfrac{\mathrm d^2y}{\mathrm dx^2}+\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2+2\tan y\,\dfrac{\mathrm dy}{\mathrm dx}=1+\mathrm e^{-2x}\sec y\).
(i) Show that \(\dfrac{\mathrm d^2w}{\mathrm dx^2}+2\dfrac{\mathrm dw}{\mathrm dx}+w=-\mathrm e^{-2x}\).
(ii) Find the particular solution for \(y\) in terms of \(x\), given that when \(x=0\), \(y=\dfrac{\pi}{3}\) and \(\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{1}{\sqrt3}\).
OR
The curves \(C_1\) and \(C_2\) have polar equations, for \(0\leq\theta\leq\dfrac{\pi}{2}\), as follows:
\(C_1: r=2\left(\mathrm e^{\theta}+\mathrm e^{-\theta}\right)\),
\(C_2: r=\mathrm e^{2\theta}-\mathrm e^{-2\theta}\).
The curves intersect at the point \(P\) where \(\theta=\alpha\).
(i) Show that \(\mathrm e^{2\alpha}-2\mathrm e^\alpha-1=0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt2\).
(ii) Sketch \(C_1\) and \(C_2\) on the same diagram.
(iii) Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures.