Matrix groups

7 The set \(M\) consists of the six matrices \(\left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\), where \(n \in \{ 0,1,2,3,4,5 \}\). It is given that \(M\) forms a group ( \(M , \times\) ) under matrix multiplication, with numerical addition and multiplication both being carried out modulo 6 .

1. Determine whether ( \(M , \times\) ) is a commutative group, justifying your answer.
2. Write down the identity element of the group and find the inverse of \(\left( \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right)\).
3. State the order of \(\left( \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right)\) and give a reason why \(( M , \times )\) has no subgroup of order 4.
4. The multiplicative group \(G\) has order 6. All the elements of \(G\), apart from the identity, have order 2 or 3 . Determine whether \(G\) is isomorphic to ( \(M , \times\) ), justifying your answer.

4 The group \(G\) consists of a set of six matrices under matrix multiplication. Two of the elements of \(G\) are \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & - 1 \\ 0 & - 1 \end{array} \right)\).

1. Determine each of the following:
   • \(\mathbf { A } ^ { 2 }\)
   • \(\mathbf { B } ^ { 2 }\)
   • Determine all the elements of \(G\).
   • State the order of each non-identity element of \(G\).
   • State, with justification, whether \(G\) is
   • abelian
   • cyclic.

1. The set of matrices \(G = \{ \mathbf { I } , \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } , \mathbf { E } \}\) where

$$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right) \quad \mathbf { D } = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right) \quad \mathbf { E } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 1 \end{array} \right)$$ with the operation \(\otimes _ { 2 }\) of matrix multiplication with entries evaluated modulo 2 , forms a group.

1. Show that \(\mathbf { B }\) is an element of order 3 in \(G\).
2. Determine the orders of the other elements of \(G\).
3. Give a reason why \(G\) is not isomorphic to
   1. a cyclic group of order 6
   2. the group of symmetries of a regular hexagon. The group \(H\) of permutations of the numbers 1, 2 and 3 contains the following elements, denoted in two-line notation, $$\begin{array} { l l l } e = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 1 & 2 & 3 \end{array} \right) & a = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 3 & 1 \end{array} \right) & b = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 3 & 1 & 2 \end{array} \right) \\ c = \left( \begin{array} { l l } 1 & 2 \\ 1 & 3 \\ 2 \end{array} \right) & d = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \end{array} \right) & f = \left( \begin{array} { l l } 1 & 2 \\ 3 & 2 \end{array} \right) \end{array}$$
4. Determine an isomorphism between the groups \(G\) and \(H\).

5 The set \(S\) consists of all \(2 \times 2\) matrices having determinant 1 or - 1 . For instance, the matrices \(\mathbf { P } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right) , \mathbf { Q } = - \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \\ - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\) are elements of \(S\). It is given that \(\times _ { \mathbf { M } }\) is the operation of matrix multiplication.

1. State the identity element of \(S\) under \(\times _ { \mathbf { M } }\). The group \(G\) is generated by \(\mathbf { P }\), under \(\times _ { \mathbf { M } }\).
2. Determine the order of \(G\). The group \(H\) is generated by \(\mathbf { Q }\) and \(\mathbf { R }\), also under \(\times _ { \mathbf { M } }\).
3. 1. By finding each element of \(H\), determine the order of \(H\).
   2. List all the proper subgroups of \(H\).
4. State whether each of the following statements is true or false. Give a reason for each of your answers.

6

1. The set \(S\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n \in \mathbb { Z }\).
   1. Show that \(S\), under the operation of matrix multiplication, forms a group \(G\). [You may assume that matrix multiplication is associative.]
   2. State, giving a reason, whether \(G\) is abelian.
   3. The group \(H\) is the set \(\mathbb { Z }\) together with the operation of addition. Explain why \(G\) is isomorphic to \(H\).
   4. The plane transformation \(T\) is given by the matrix \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n\) is a non-zero integer. Describe \(T\) fully.

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where \(p\) is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.

11

1. (a) Given \(\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)\), work out the matrix \(\mathbf { A B }\) and write down expressions for \(\operatorname { det } \mathbf { A }\) and \(\operatorname { det } \mathbf { B }\).
   (b) Verify, by direct calculation, that \(\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }\). Let \(S\) be the set of all \(2 \times 2\) matrices with determinant equal to 1 .
2. Show that \(\left( S , \times _ { \mathrm { M } } \right)\) forms a group, \(G\), where \(\times _ { \mathrm { M } }\) is the operation of matrix multiplication. [You may assume that \(\mathrm { X } _ { \mathrm { M } }\) is associative.]
3. (a) Show that \(\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)\) is an element of \(G\). Let \(H\) be the smallest subgroup of \(G\) that contains \(\mathbf { K }\) and let \(n\) be the order of \(H\).
   (b) Determine the value of \(n\).
   (c) Give a second subgroup of \(G\), also of order \(n\), which is isomorphic to \(H\).

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3 .

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.

A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.

1. Write down the elements of a subgroup of order 2. [1]
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
4. Find the inverses of \(B\) and \(E\). [3]
5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]

A class of students is set the task of finding a group of functions, under composition of functions, of order 6. Student P suggests that this can be achieved by finding a function \(f\) for which \(f^6(x) = x\) and using this as a generator for the group.

1. Explain why the suggestion by Student P might not work. [2]

Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), under the operation of matrix multiplication.

1. Explain why such a group is only possible if \(\det(\mathbf{M}) = 1\) or \(-1\). [2]
2. Write down values of \(a\), \(b\), \(c\) and \(d\) that would give a suitable matrix \(\mathbf{M}\) for which \(\mathbf{M}^6 = \mathbf{I}\) and \(\det(\mathbf{M}) = 1\). [1]

Student Q believes that it is possible to construct a rational function \(f\) in the form \(f(x) = \frac{ax + b}{cx + d}\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf{M}\) of part (iii).

1. 1. Write down and simplify the function \(f\) that, according to Student Q, corresponds to \(\mathbf{M}\). [1]
   2. By calculating \(\mathbf{M}^2\), show that Student Q's suggestion does not work. [2]
   3. Find a different function \(f\) that will satisfy the requirements of the task. [4]

The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\begin{pmatrix} x & -y \\ y & x \end{pmatrix}\), where \(x\) and \(y\) are real numbers which are not both zero.

1. 1. The matrices \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\) and \(\begin{pmatrix} c & -d \\ d & c \end{pmatrix}\) are both elements of \(X\). Show that \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\begin{pmatrix} c & -d \\ d & c \end{pmatrix} = \begin{pmatrix} p & -q \\ q & p \end{pmatrix}\) for some real numbers \(p\) and \(q\) to be found in terms of \(a\), \(b\), \(c\) and \(d\). [2]
   2. Prove by contradiction that \(p\) and \(q\) are not both zero. [5]
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.] [4]
3. Determine a subgroup of \(G\) of order 17. [2]

Consider the set \(S\) of all matrices of the form \(\begin{pmatrix} p & p \\ p & p \end{pmatrix}\), where \(p\) is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). [5]
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3. [4]