SL Paper 2
Consider the matrix
A \( = \left[ {\begin{array}{*{20}{c}} \lambda &3&2 \\ 2&4&\lambda \\ 3&7&3 \end{array}} \right]\).
Suppose now that \(\lambda = 1\) so consider the matrix
B \( = \left[ {\begin{array}{*{20}{c}} 1&3&2 \\ 2&4&1 \\ 3&7&3 \end{array}} \right]\).
Find an expression for det(A) in terms of \(\lambda \), simplifying your answer.
Hence show that A is singular when \(\lambda = 1\) and find the other value of \(\lambda \) for which A is singular.
Explain how it can be seen immediately that B is singular without calculating its determinant.
Determine the null space of B.
Explain briefly how your results verify the rank-nullity theorem.
Prove, using mathematical induction, that
B\(^n = {8^{n - 2}}\)B\(^2\) for \(n \in {\mathbb{Z}^ + },{\text{ }}n \geqslant 3\).
The hyperbola with equation \({x^2} - 4xy - 2{y^2} = 3\) is rotated through an acute anticlockwise angle \(\alpha \) about the origin.
The point \((x,{\text{ }}y)\) is rotated through an anticlockwise angle \(\alpha \) about the origin to become the point \((X,{\text{ }}Y)\). Assume that the rotation can be represented by
\[\left[ {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right].\]
Show, by considering the images of the points \((1,{\text{ }}0)\) and \((0,{\text{ }}1)\) under this rotation that
\[\left[ {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} {\cos \alpha }&{ - \sin \alpha } \\ {\sin \alpha }&{\cos \alpha } \end{array}} \right].\]
By expressing \((x,{\text{ }}y)\) in terms of \((X,{\text{ }}Y)\), determine the equation of the rotated hyperbola in terms of \(X\) and \(Y\).
Verify that the coefficient of \(XY\) in the equation is zero when \(\tan \alpha = \frac{1}{2}\).
Determine the equation of the rotated hyperbola in this case, giving your answer in the form \(\frac{{{X^2}}}{{{A^2}}} - \frac{{{Y^2}}}{{{B^2}}} = 1\).
Hence find the coordinates of the foci of the hyperbola prior to rotation.
\(S\) is defined as the set of all \(2 \times 2\) non-singular matrices. \(A\) and \(B\) are two elements of the set \(S\).
(i) Show that \({({A^T})^{ - 1}} = {({A^{ - 1}})^T}\).
(ii) Show that \({(AB)^T} = {B^T}{A^T}\).
A relation \(R\) is defined on \(S\) such that \(A\) is related to \(B\) if and only if there exists an element \(X\) of \(S\) such that \(XA{X^T} = B\). Show that \(R\) is an equivalence relation.
The matrix A is given by A = \(\left( {\begin{array}{*{20}{c}}1&2&3&4\\3&8&{11}&8\\1&3&4&\lambda \\\lambda &5&7&6\end{array}} \right)\).
(a) Given that \(\lambda = 2\), B = \(\left( \begin{array}{l}2\\4\\\mu \\3\end{array} \right)\) and X = \(\left( \begin{array}{l}x\\y\\z\\t\end{array} \right)\),
(i) find the value of \(\mu \) for which the equations defined by AX = B are consistent and solve the equations in this case;
(ii) define the rank of a matrix and state the rank of A.
(b) Given that \(\lambda = 1\),
(i) show that the four column vectors in A form a basis for the space of four-dimensional column vectors;
(ii) express the vector \(\left( \begin{array}{c}6\\28\\12\\15\end{array} \right)\) as a linear combination of these basis vectors.
The set of all permutations of the list of the integers \(1,{\text{ }}2,{\text{ }}3{\text{ }} \ldots {\text{ }}n\) is a group, \({S_n}\), under the operation of composition of permutations.
Each element of \({S_4}\) can be represented by a \(4 \times 4\) matrix. For example, the cycle \({\text{(1 2 3 4)}}\) is represented by the matrix
\(\left( {\begin{array}{*{20}{c}} 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \\ 1&0&0&0 \end{array}} \right)\) acting on the column vector \(\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \\ 4 \end{array}} \right)\).
(i) Show that the order of \({S_n}\) is \(n!\);
(ii) List the 6 elements of \({S_3}\) in cycle form;
(iii) Show that \({S_3}\) is not Abelian;
(iv) Deduce that \({S_n}\) is not Abelian for \(n \geqslant 3\).
(i) Write down the matrices M\(_1\), M\(_2\) representing the permutations \((1{\text{ }}2),{\text{ }}(2{\text{ }}3)\), respectively;
(ii) Find M\(_1\)M\(_2\) and state the permutation represented by this matrix;
(iii) Find \(\det (\)M\(_1)\), \(\det (\)M\(_2)\) and deduce the value of \(\det (\)M\(_1\)M\(_2)\).
(i) Use mathematical induction to prove that
\((1{\text{ }}n)(1{\text{ }}n{\text{ }} - 1)(1{\text{ }}n - 2) \ldots (1{\text{ }}2) = (1{\text{ }}2{\text{ }}3 \ldots n){\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}n > 1\).
(ii) Deduce that every permutation can be written as a product of cycles of length 2.
Given that the elements of a \(2 \times 2\) symmetric matrix are real, show that
(i) the eigenvalues are real;
(ii) the eigenvectors are orthogonal if the eigenvalues are distinct.
The matrix \(\boldsymbol{A}\) is given by\[\boldsymbol{A} = \left( {\begin{array}{*{20}{c}}
{11}&{\sqrt 3 }\\
{\sqrt 3 }&9
\end{array}} \right) .\]Find the eigenvalues and eigenvectors of \(\boldsymbol{A}\).
The ellipse \(E\) has equation \({{\boldsymbol{X}}^T}{\boldsymbol{AX}} = 24\) where \(\boldsymbol{X} = \left( \begin{array}{l}
x\\
y
\end{array} \right)\) and \(\boldsymbol{A}\) is as defined in part (b).
(i) Show that \(E\) can be rotated about the origin onto the ellipse \(E'\) having equation \(2{x^2} + 3{y^2} = 6\) .
(ii) Find the acute angle through which \(E\) has to be rotated to coincide with \(E'\) .
The function \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) is defined by \(\boldsymbol{X} \mapsto \boldsymbol{AX}\) , where \(\boldsymbol{X} = \left[ \begin{array}{l}
x\\
y
\end{array} \right]\) and \(\boldsymbol{A} = \left[ \begin{array}{l}
a\\
c
\end{array} \right.\left. \begin{array}{l}
b\\
d
\end{array} \right]\) where \(a\) , \(b\) , \(c\) , \(d\) are all non-zero.
Consider the group \(\left\{ {S,{ + _m}} \right\}\) where \(S = \left\{ {0,1,2 \ldots m - 1} \right\}\) , \(m \in \mathbb{N}\) , \(m \ge 3\) and \({ + _m}\) denotes addition modulo \(m\) .
Show that \(f\) is a bijection if \(\boldsymbol{A}\) is non-singular.
Suppose now that \(\boldsymbol{A}\) is singular.
(i) Write down the relationship between \(a\) , \(b\) , \(c\) , \(d\) .
(ii) Deduce that the second row of \(\boldsymbol{A}\) is a multiple of the first row of \(\boldsymbol{A}\) .
(iii) Hence show that \(f\) is not a bijection.
Show that \(\left\{ {S,{ + _m}} \right\}\) is cyclic for all m .
Given that \(m\) is prime,
(i) explain why all elements except the identity are generators of \(\left\{ {S,{ + _m}} \right\}\) ;
(ii) find the inverse of \(x\) , where x is any element of \(\left\{ {S,{ + _m}} \right\}\) apart from the identity;
(iii) determine the number of sets of two distinct elements where each element is the inverse of the other.
Suppose now that \(m = ab\) where \(a\) , \(b\) are unequal prime numbers. Show that \(\left\{ {S,{ + _m}} \right\}\) has two proper subgroups and identify them.
By considering the points \((1,{\text{ }}0)\) and \((0,{\text{ }}1)\) determine the \(2 \times 2\) matrix which represents
(i) an anticlockwise rotation of \(\theta \) about the origin;
(ii) a reflection in the line \(y = (\tan \theta )x\).
Determine the matrix \(A\) which represents a rotation from the direction \(\left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right)\) to the direction \(\left( {\begin{array}{*{20}{c}} 1 \\ 3 \end{array}} \right)\).
A triangle whose vertices have coordinates \((0,{\text{ }}0)\), \((3,{\text{ }}1)\) and \((1,{\text{ }}5)\) undergoes a transformation represented by the matrix \({A^{ - 1}}XA\), where \(X\) is the matrix representing a reflection in the \(x\)-axis. Find the coordinates of the vertices of the transformed triangle.
The matrix \(B = {A^{ - 1}}XA\) represents a reflection in the line \(y = mx\). Find the value of \(m\).