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Section 8.3 Excursion: Matrices and Finite Groups

Matrices (arrays) are a common discrete structure. They are used to store information of all kinds, including graphs, as we saw in Section 3.2. In this section, we learn how to multiply \(2 \times 2\) matrices and look for patterns in the powers of \(2 \times 2\) matrices. We verify our conjectures using mathematical induction. Finally, we look at the abstract concept of a group.

Subsection 8.3.A Matrix Products and Powers

We begin with some definitions.

Definition 8.3.1 Matrix multiplication and powers.

 
  1. A \(m \times n\) matrix is an array with entries in each of \(m\) rows and \(n\) columns. For example,
    \begin{equation*} C = \left[ \begin{array}{ccc} -2 \amp 3 \amp 0 \\ -1 \amp 1 \amp 2 \end{array} \right] \end{equation*}
    is a \(2 \times 3\) matrix.
  2. In this textbook, we only work with square matrices where the number of rows equals the number of columns. For example,
    \begin{equation*} B = \left[ \begin{array}{rr} 2 \amp -3 \\ 1 \amp 2 \end{array} \right] \end{equation*}
    is a square \(2 \times 2\) matrix.
  3. Two special \(2 \times 2\) matrices are the zero matrix \(\mathbf{0} = \left[ \begin{array}{cc} 0 \amp 0\\ 0 \amp 0 \end{array} \right]\) and the identity matrix \(I = \left[ \begin{array}{cc} 1 \amp 0\\ 0 \amp 1 \end{array} \right]\text{.}\)
  4. For \(2 \times 2\) matrices, the complicated formula for matrix multiplication is
    \begin{equation*} \left[ \begin{array}{cc} a_1 \amp b_1 \\ c_1 \amp d_1 \end{array} \right]\left[ \begin{array}{cc} a_2 \amp b_2 \\ c_2 \amp d_2 \end{array} \right]=\left[ \begin{array}{cc} a_1a_2 + b_1c_2 \amp a_1b_2+b_1d_2\\ c_1a_2 + d_1c_2 \amp c_1b_2+d_1d_2 \end{array} \right]\text{.} \end{equation*}
    To obtain the entry in the \(i\)th row and the \(j\)th column of the matrix product, we use the \(i\)th row of the first matrix and the \(j\)th column of the second matrix. For example, the entry in the first row and second column of the matrix product is \(a_1b_2+b_1d_2\) which uses the first row (\(a_1\) and \(b_1\)) of the first matrix and the second column (\(b_2\) and \(d_2\)) of the second matrix. For example, consider the matrices \(A=\left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 3 \end{array} \right]\) and \(B=\left[ \begin{array}{rr} 2 \amp -3 \\ 1 \amp 2 \end{array} \right]\text{.}\) Then
    \begin{equation*} AB=\left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 3 \end{array} \right]\left[ \begin{array}{rr} 2 \amp -3 \\ 1 \amp 2 \end{array} \right] =\left[ \begin{array}{cc} (-1)(2)+(2)(1) \amp (-1)(-3)+(2)(2)\\ (0)(2)+(3)(1) \amp (0)(-3)+(3)(2) \end{array} \right] = \left[ \begin{array}{rr} 0 \amp 7 \\ 3 \amp 6 \end{array} \right]\text{.} \end{equation*}
  5. Matrix multiplication is associative
    \begin{equation*} A(BC)=(AB)C \end{equation*}
    but it is not commutative. For example, for those same matrices \(A\) and \(B\) we have
    \begin{equation*} BA = \left[ \begin{array}{rr} 2 \amp -3 \\ 1 \amp 2 \end{array} \right]\left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 3 \end{array} \right] =\left[ \begin{array}{cc} (2)(-1)+(-3)(0) \amp (2)(2)+(-3)(3))\\ (1)(-1)+(2)(0) \amp (1)(2)+(2)(3) \end{array} \right] = \left[ \begin{array}{rr} -2 \amp -5 \\ -1 \amp 8 \end{array} \right] \end{equation*}
  6. For a square matrix \(A\) and a positive integer \(s\text{,}\) the matrix power is
    \begin{equation*} A^s= \underbrace{AA\cdots A}_{s \text{ times } } \end{equation*}
    For example,
    \begin{equation*} \left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 3 \end{array} \right]^2 = \left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 3 \end{array} \right]\left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 3 \end{array} \right] =\left[ \begin{array}{cc} (-1)(-1)+2(0) \amp (-1)(2)+(2)(3)\\ (0)(-1)+(3)(0) \amp (0)(2)+(3)(3) \end{array} \right] = \left[ \begin{array}{rr} 1 \amp 4 \\ 0 \amp 9 \end{array} \right]\text{.} \end{equation*}
In our next example, we look at higher powers of a matrix.

Example 8.3.2 Pattern in matrix powers.

Consider the matrix \(A = \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\text{.}\)
  1. Calculate \(A^2\) and \(A^3\text{.}\)
    Solution.
    We have
    \begin{equation*} A^2 = \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right] \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right] = \left[ \begin{array}{cc} 1+0 \amp 1+1\\ 0+0 \amp 0+1 \end{array} \right] = \left[ \begin{array}{cc} 1 \amp 2\\ 0 \amp 1 \end{array} \right]\text{.} \end{equation*}
    Next,
    \begin{equation*} A^3 = AA^2= \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right] \left[ \begin{array}{cc} 1 \amp 2\\ 0 \amp 1 \end{array} \right] = \left[ \begin{array}{cc} 1+0 \amp 2+1\\ 0+0 \amp 0+1 \end{array} \right] = \left[ \begin{array}{cc} 1 \amp 3\\ 0 \amp 1 \end{array} \right] \end{equation*}
  2. Make a conjecture about an explicit formula for \(A^n\) for \(n \ge 1\text{.}\)
    Solution.
    Based on our examples, we might conjecture that \(A^n=\left[ \begin{array}{cc} 1 \amp n\\ 0 \amp 1 \end{array} \right]\text{.}\)
Let’s practice calculating matrix products and powers.

Activity 8.3.A Matrix multiplication and powers.

  Consider the matrix \(A=\left[ \begin{array}{cc} 1 \amp 0 \\ 3 \amp 1 \end{array} \right]\text{.}\)
  1. Calculate \(A^2\text{.}\)
  2. Calculate \(A^3\) using that \(A^3=AA^2\text{.}\)
  3. Conjecture an explicit formula for \(A^n\text{.}\)
We can define functions that involve matrices.

Example 8.3.4 Functions that involve matrices.

Let \(\mathscr{M}(\mathbb{Z})\) be the set of \(2 \times 2\) matrices with integer entries.
  1. Consider the function \(f: \mathbb{Z} \to \mathscr{M}(\mathbb{Z})\) defined by \(f(n) = \left[ \begin{array}{cc} 1 \amp 0 \\ 3n \amp 1 \end{array} \right]\text{.}\) Identify the domain and codomain of \(f\) and evaluate \(f(2)\text{.}\)
    Solution.
    The domain is \(\mathbb{Z}\text{,}\) the codomain is \(\mathscr{M}(\mathbb{Z})\text{,}\) and \(f(2) = \left[ \begin{array}{cc} 1 \amp 0 \\ 6 \amp 1 \end{array} \right]\text{.}\)
  2. Consider the function \(\fn{det}: \mathscr{M}(\mathbb{Z}) \to \mathbb{Z}\) defined by \(\fn{det} \left( \left[ \begin{array}{cc} a \amp b \\ c \amp d \end{array} \right] \right) = ad-bc\text{.}\) Identify the domain and codomain of \(\fn{det}\) and evaluate \(\fn{det}\left( \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] \right)\text{.}\)
    Solution.
    The domain is \(\mathscr{M}(\mathbb{Z})\text{,}\) the codomain is \(\mathbb{Z}\text{,}\) and \(\fn{det}\left( \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] \right)=(1)(5)-(2)(3) = -1\text{.}\)
  3. Consider the function \(T: \mathscr{M}(\mathbb{Z}) \to \mathscr{M}(\mathbb{Z})\) defined by \(T(A) = MA\) where \(M = \left[ \begin{array}{cr} 1 \amp 2 \\ 0 \amp -1 \end{array} \right]\text{.}\) Identify the domain and codomain of \(T\) and evaluate \(T\left( \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] \right)\text{.}\)
    Solution.
    The domain is \(\mathscr{M}(\mathbb{Z})\text{,}\) the codomain is also \(\mathscr{M}(\mathbb{Z})\text{,}\) and
    \begin{equation*} T\left( \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] \right) = \left[ \begin{array}{cr} 1 \amp 2 \\ 0 \amp -1 \end{array} \right] \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] = \left[ \begin{array}{cc} 1+6 \amp 2+10 \\ 0-3 \amp 0-5 \end{array} \right] = \left[ \begin{array}{rr} 7 \amp 12 \\ -3 \amp -5 \end{array} \right] \end{equation*}
We can also define sets that involve matrices.

Example 8.3.5 Sets that involve matrices.

 
  1. Give two examples of elements in the set
    \begin{equation*} \left\{\left[ \begin{array}{cc} a \amp b \\ 0 \amp a \end{array} \right] \mid a, b \in \mathbb{Z} \right\} \end{equation*}
    Solution.
    We can choose any integer values of \(a\) and \(b\text{.}\) For example, when \(a = 2\) and \(b=3\text{,}\) we get the element \(\left[ \begin{array}{cc} 2 \amp 3 \\ 0 \amp 2 \end{array} \right]\text{.}\) As another example, when \(a = 1\) and \(b=0\text{,}\) we get the element \(I\text{.}\)
  2. Confirm that \(I\) and \(C = \left[ \begin{array}{cc} 0 \amp -1 \\ -1 \amp 0 \end{array} \right]\) are elements of the set \(\mathscr{B} = \left\{ B \in \mathscr{M}(\mathbb{Z}) \mid B^2=I \right\}\)
    Solution.
    We check that
    \begin{equation*} I^2 = \left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 1 \end{array} \right] \left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 1 \end{array} \right] = \left[ \begin{array}{cc} 1+0 \amp 0+0 \\ 0+0 \amp 0+1 \end{array} \right] = \left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 1 \end{array} \right] = I\text{.} \end{equation*}
    Thus \(I \in \mathscr{B}\text{.}\) Next, we check that
    \begin{equation*} C^2= \left[ \begin{array}{cc} 0 \amp -1 \\ -1 \amp 0 \end{array} \right] \left[ \begin{array}{cc} 0 \amp -1 \\ -1 \amp 0 \end{array} \right] = \left[ \begin{array}{cc} 0+1 \amp 0+0 \\ 0+0 \amp 1+0 \end{array} \right] = \left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 1 \end{array} \right] = I\text{.} \end{equation*}
    Thus \(C \in \mathscr{B}\text{.}\)
  3. Give an explicit definition of the set
    \begin{equation*} \mathscr{A}= \{ A \in \mathscr{M}(\mathbb{Z}) \mid NA = \mathbf{0}\} \end{equation*}
    where \(N = \left[ \begin{array}{cc} 1 \amp 2 \\ 1 \amp 2 \end{array} \right]\text{.}\)
    Solution.
    Suppose \(A = \left[ \begin{array}{cc} a \amp b \\ c \amp d \end{array} \right] \in \mathscr{A}\text{.}\) Then
    \begin{equation*} NA = \left[ \begin{array}{cc} 1 \amp 2 \\ 1 \amp 2 \end{array} \right] \left[ \begin{array}{cc} a \amp b \\ c \amp d \end{array} \right] = \left[ \begin{array}{cc} a+2c \amp b+2d \\ a+2c \amp b+2d \end{array} \right] = \left[ \begin{array}{cc} 0 \amp 0 \\ 0 \amp 0 \end{array} \right]\text{.} \end{equation*}
    We must have \(a+2c = 0\) which means \(a = -2c\text{.}\) We must also have \(b+2d = 0\) which means \(b = -2d\text{.}\) That is,
    \begin{equation*} A = \left[ \begin{array}{rr} -2c \amp -2d \\ c \amp d \end{array} \right]\text{.} \end{equation*}
    We can write
    \begin{equation*} \mathscr{A} = \left\{ \left[ \begin{array}{rr} -2c \amp -2d \\ c \amp d \end{array} \right] \mid c, d \in \mathbb{Z} \right\} \end{equation*}

Subsection 8.3.B Proof Format: Special Case of Mathematical Induction for Proving an Explicit Formula for the Power of a Matrix

We can use mathematical induction to verify our conjectures about powers of matrices.
Try using this new proof format.

Activity 8.3.B Matrix power proof by mathematical induction.

 
  1. For the matrix \(A = \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\text{,}\) in Example 8.3.2 we conjectured that \(A^n=\left[ \begin{array}{cc} 1 \amp n\\ 0 \amp 1 \end{array} \right]\) for \(n \ge 1\text{.}\) Copy the following proof by mathematical induction justifying our conjecture.

    Proof.
    Write \(M_n=\left[ \begin{array}{cc} 1 \amp n\\ 0 \amp 1 \end{array} \right]\text{.}\) Induct on \(n\text{.}\)
    First, when \(n=1\) we have \(A^1 = A = \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\) and also \(M_1=\left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\text{.}\)
    Next, assume \(A^{n-1} = M_{n-1} = \left[ \begin{array}{cc} 1 \amp n-1 \\ 0 \amp 1 \end{array} \right]\) for some integer \(n \ge 2\text{.}\) Then
    \begin{equation*} A^n = AA^{n-1} =AM_{n-1} = \left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\left[ \begin{array}{cc} 1 \amp n-1\\ 0 \amp 1 \end{array} \right] = \left[ \begin{array}{cc} 1+0 \amp (n-1)+1\\ 0+0 \amp 0+1 \end{array} \right] = \left[ \begin{array}{cc} 1 \amp n\\ 0 \amp 1 \end{array} \right]=M_n \end{equation*}
  2. What was the inductive hypothesis (part we assumed)?
  3. Explain the algebra step-by-step in the last line of the proof.
  4. For the matrix \(A=\left[ \begin{array}{cc} 1 \amp 0 \\ 3 \amp 1 \end{array} \right]\text{,}\) in Activity 8.3.A we conjectured that
    \begin{equation*} A^n=\left[ \begin{array}{cc} 1 \amp 0 \\ 3n \amp 1 \end{array} \right] \end{equation*}
    for \(n \ge 1\text{.}\) Prove this conjecture using mathematical induction by changing what needs to be changed in the proof form (a). Caution: When you are evaluating \(M_{n-1}\) be sure to put \((n-1)\) in parentheses.

Subsection 8.3.C Finite Groups

We have seen several examples of finite sets with an operation. For example, in Section 4.3, we saw \(\mathbb{Z}_n\text{,}\) the integers mod \(n\) with the operations of addition and multiplication. In Section 6.4, we saw permutations with the operation of function composition. And now we have seen \(2 \times 2\) matrices with matrix multiplication. These structures have several common properties. Let’s look at one abstraction, a finite group. Here is the definition.

Definition 8.3.8 Group.

Let \(G\) be a set with an operation \(\star\) such that
  1. \(G\) is closed under \(\star\text{:}\) \(a \star b \in G\) for all elements \(a, b \in G\text{.}\)
  2. \(\star\) is associative: \(a \star (b \star c) = (a \star b) \star c\) for all elements \(a, b,c \in G\text{.}\)
  3. There is an identity element \(e \in G\) such that \(a \star e = a\) and \(e \star a = a\) for all elements \(a \in G\text{.}\)
  4. For each element \(a \in G\text{,}\) there exists an inverse element \(b \in G\) such that \(a \star b = e\) and \(b \star a = e\text{.}\)
Let’s look at a few examples.

Example 8.3.9 \(\mathbb{Z}_4\) is a group under addition.

In Section 4.3 we saw that the integers mod 4 is the set \(\mathbb{Z}_4 =\{0,1,2,3\}\) with addition defined by \(a \oplus b = (a+b) \fn{ mod } 4\text{.}\)
  1. Write the addition table for \(\mathbb{Z}_4\text{.}\)
    Solution.
    We show the addition table in Table 8.3.10. For example, \(2 \oplus 3 = 5 \fn{ mod } 4 = 1\text{.}\)
    Table 8.3.10. The addition table \(\mathbb{Z}_4\)
    \(\oplus\) 0 1 2 3
    0 0 1 2 3
    1 1 2 3 0
    2 2 3 0 1
    3 3 0 1 2
  2. Check that \(\mathbb{Z}_4\) is a group under addition.
    Solution.
    First, every entry in table is 0, 1, 2, or 3 so \(\mathbb{Z}_4\) is closed under \(\oplus\text{.}\) Next, \(\oplus\) is associative because
    \begin{equation*} a \oplus (b \oplus c) = (a+(b+c)) \fn{ mod } 4 = ((a+b)+c) \fn{ mod } 4 = (a \oplus b) \oplus c \end{equation*}
    where we used that addition in \(\mathbb{Z}\) is associative. The identity element is 0, as you can check in the table. Note that \(0\oplus0 = 0\text{,}\) \(1 \oplus 3 = 0\text{,}\) \(2 \oplus 2 = 0\text{,}\) and \(3 \oplus 1 = 0\text{.}\) Therefore, the inverse of 0 is 0, the inverse of 1 is 3, the inverse of 2 is 2, and the inverse of 3 is 1. Sometimes we say the additive identity or the additive inverse to emphasis the operation is addition.
Here is an example that involves multiplication.

Example 8.3.11 Finite multiplicative group mod \(12\).

Recall that the integers mod 12 are \(\mathbb{Z}_{12} =\{0,1,2,\ldots, 11\}\text{.}\) Multiplication is defined by \(a \otimes b = (ab) \fn{ mod } 12\text{.}\) Consider the subset \(\mathbb{U}_{12} = \{1, 5, 7, 11\}\) of \(\mathbb{Z}_{12}\text{.}\)
  1. Write the multiplication table for \(\mathbb{U}_{12}\text{.}\)
    Solution.
    We show the multiplication table in Table 8.3.12. For example, \(5 \oplus 7 = 35 \fn{ mod } 12 = 11\) because \(35-12-12=11\text{.}\)
    Table 8.3.12. The multiplication table for \(\mathbb{U}_{12}\)
    \(\otimes\) 1 5 7 11
    1 1 5 7 11
    5 5 1 11 7
    7 7 11 1 5
    11 11 7 5 1
  2. Check that \(\mathbb{U}_{12}\) is a group under multiplication.
    Solution.
    First, every entry in table is 1, 5, 7, or 1 so \(\mathbb{U}_{12}\) is closed under \(\otimes\text{.}\) Next, \(\otimes\) is associative because
    \begin{equation*} a \otimes (b \otimes c) = (a(bc)) \fn{ mod } 12 = ((ab)c) \fn{ mod } 12 = (a \otimes b) \otimes c \end{equation*}
    where we used that multiplication in \(\mathbb{Z}\) is associative. The identity element is 1, as you can check in the table. Note that \(1\otimes 1 = 1\text{,}\) \(5 \otimes 5 = 1\text{,}\) \(7 \otimes 7 = 1\text{,}\) and \(11 \otimes 11 = 1\text{.}\) Therefore, the inverse of 1 is 1, the inverse of 5 is 5, the inverse of 7 is 7, and the inverse of 11 is 11. Sometimes we say the multiplicative identity or the multiplicative inverse to emphasis the operation is multiplication.
The groups in Example 8.3.9 and Example 8.3.11 have an additional property.

Definition 8.3.13 Commutative group.

A group \(G\) with operation \(\star\) is abelian (or commutative) if \(a \star b = b \star a\) for all elements \(a, b \in G\text{.}\) For example, \(\mathbb{Z}_4\) and \(\mathbb{U}_{12}\) are abelian.
Let’s look at a group that is not abelia.

Example 8.3.14 Nonabelian group.

Let \(\mathbb{S}_3\) be the set of all permutations on the set \(\{1,2,3\}\) with the operation \(\circ\) which is function composition, introduced in Section 6.4. Write
\begin{equation*} \iota = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 1 \amp 2 \amp 3 \end{array} \right) \sigma_1 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 2 \amp 3 \amp 1 \end{array} \right) \sigma_2 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 3 \amp 1 \amp 2 \end{array} \right) \end{equation*}
\begin{equation*} \tau_1 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 1 \amp 3 \amp 2 \end{array} \right) \tau_2 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 3 \amp 2 \amp 1 \end{array} \right) \tau_3 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 2 \amp 1 \amp 3 \end{array} \right) \end{equation*}
  1. Write the operation table for \(\mathbb{S}_3 = \{ \iota, \sigma_1, \sigma_2, \tau_1, \tau_2, \tau_3\}\text{.}\)
    Solution.
    We show the operation table in Table 8.3.16. For example,
    \begin{equation*} \tau_1 \circ \sigma_1 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 1 \amp 3 \amp 2 \end{array} \right) \circ \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 2 \amp 3 \amp 1 \end{array} \right) = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 3 \amp 2 \amp 1 \end{array} \right)= \tau_2\text{.} \end{equation*}
    Recall that we can compose permutations using dot diagrams as in Figure 8.3.15.
    Figure 8.3.15. Calculating the composite of permutations \(\tau_1 \circ \sigma_1\) in \(\mathbb{S}_3\) using dot diagrams.
    Be careful as
    \begin{equation*} \sigma_1 \circ \tau_1 = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 2 \amp 3 \amp 1 \end{array} \right) \circ \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 1 \amp 3 \amp 2 \end{array} \right) = \left( \begin{array}{ccc} 1 \amp 2 \amp 3 \\ 2 \amp 1 \amp 3 \end{array} \right)= \tau_3\text{.} \end{equation*}
    Table 8.3.16. The operation table for \(\mathbb{S}_3\)
    \(\circ\) \(\iota\) \(\sigma_1\) \(\sigma_2\) \(\tau_1\) \(\tau_2\) \(\tau_3\)
    \(\iota\) \(\iota\) \(\sigma_1\) \(\sigma_2\) \(\tau_1\) \(\tau_2\) \(\tau_3\)
    \(\sigma_1\) \(\sigma_1\) \(\sigma_2\) \(\iota\) \(\tau_3\) \(\tau_1\) \(\tau_2\)
    \(\sigma_2\) \(\sigma_2\) \(\iota\) \(\sigma_1\) \(\tau_2\) \(\tau_3\) \(\tau_1\)
    \(\tau_1\) \(\tau_1\) \(\tau_2\) \(\tau_3\) \(\iota\) \(\sigma_1\) \(\sigma_2\)
    \(\tau_2\) \(\tau_2\) \(\tau_3\) \(\tau_1\) \(\sigma_2\) \(\iota\) \(\sigma_1\)
    \(\tau_3\) \(\tau_3\) \(\tau_1\) \(\tau_2\) \(\sigma_1\) \(\sigma_2\) \(\iota\)
  2. Check that \(\mathbb{S}_3\) is a group under function composition.
    Solution.
    First, every entry in table is \(\iota\text{,}\) \(\sigma_1\text{,}\) \(\sigma_2\text{,}\) \(\tau_1\text{,}\) \(\tau_2\text{,}\) or \(\tau_3\) so \(\mathbb{S}_3\) is closed under \(\circ\text{.}\) Next, \(\circ\) is associative because
    \begin{equation*} (\alpha \circ (\beta \circ \gamma)) (n) = \alpha((\beta \circ \gamma)(n)) = \alpha(\beta(\gamma(n))) = (\alpha \circ \beta) (\gamma(n)) = ((\alpha \circ \beta ) \circ \gamma) (n) \end{equation*}
    for \(n=1,2,3\text{.}\) Thus \(\alpha \circ (\beta \circ \gamma)= (\alpha \circ \beta ) \circ \gamma\text{.}\) The identity element is \(\iota\text{,}\) as you can check in the table. Note that the inverse of \(\iota\) is \(\iota\text{,}\) the inverse of \(\sigma_1\) is \(\sigma_2\text{,}\) the inverse of \(\sigma_2\) is \(\sigma_1\text{,}\) the inverse of \(\tau_1\) is \(\tau_1\text{,}\) the inverse of \(\tau_2\) is \(\tau_2\text{,}\) and the inverse of \(\tau_3\) is \(\tau_3\text{.}\) Sometimes we say the identity function or the inverse function to emphasis the operation is composition.
Now it is your turn to work with a finite group.

Activity 8.3.C Finite group of matrices.

Consider the following matrices
 1 
If you have studied linear algebra, the transformations corresponding to these matrices are as follows. \(I\) is the matrix of the identity transformation, \(R\) is the matrix of the rotation by 90\(^\circ\) counterclockwise, \(S\) is the matrix of the rotation by 180\(^\circ\text{,}\) \(T\) is the matrix of rotation by 90\(^\circ\) clockwise, \(H\) is the matrix of the reflection across the horizontal \(x\)-axis, \(V\) is the matrix of the reflection across the vertical \(y\)-axis, \(P\) is the matrix of the reflection across the positive slope line \(y=x\text{,}\) and \(N\) is the matrix of the reflection across the negative slope line \(y=-x\text{.}\)
:
\begin{equation*} I = \left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 1 \end{array} \right] R = \left[ \begin{array}{rr} 0 \amp -1 \\ 1 \amp 0 \end{array} \right] S = \left[ \begin{array}{rr} -1 \amp 0 \\ 0 \amp -1 \end{array} \right] T = \left[ \begin{array}{rr} 0 \amp 1 \\ -1 \amp 0 \end{array} \right] \end{equation*}
\begin{equation*} H = \left[ \begin{array}{rr} 1 \amp 0 \\ 0 \amp -1 \end{array} \right] V = \left[ \begin{array}{rr} -1 \amp 0 \\ 0 \amp 1 \end{array} \right] P = \left[ \begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \end{array} \right] N = \left[ \begin{array}{rr} 0 \amp -1 \\ -1 \amp 0 \end{array} \right] \end{equation*}
Let \(\mathbb{D} = \{I, R, S, T, H, V, P, N\}\text{.}\)
  1. Calculate each of the following products: \(RS\text{,}\) \(TV\text{,}\) \(IN\text{,}\) and \(P^2\text{.}\) If you are working with a partner, feel free to each take two of the products.
  2. The rest of the (matrix) multiplication table for \(\mathbb{D}\) shown in Table 8.3.18. There are a few extra dividing lines to help you read the table.
    Table 8.3.18. (Most of) the multiplication table for the matrix group \(\mathbb{D}\)
    × \(I\) \(R\) \(S\) \(T\) \(H\) \(V\) \(P\) \(N\)
    \(I\) \(I\) \(R\) \(S\) \(T\) \(H\) \(V\) \(P\)
    \(R\) \(R\) \(S\) \(I\) \(P\) \(N\) \(V\) \(H\)
    \(S\) \(S\) \(T\) \(I\) \(R\) \(V\) \(H\) \(N\) \(P\)
    \(T\) \(T\) \(S\) \(R\) \(I\) \(N\) \(P\) \(V\)
    \(H\) \(H\) \(N\) \(V\) \(P\) \(I\) \(S\) \(T\) \(R\)
    \(V\) \(V\) \(P\) \(H\) \(N\) \(S\) \(I\) \(R\) \(T\)
    \(P\) \(P\) \(H\) \(N\) \(V\) \(R\) \(T\) \(S\)
    \(N\) \(N\) \(V\) \(P\) \(H\) \(T\) \(R\) \(S\) \(I\)
    Check that each matrix occurs within the table once per row and once per column. If not, check your answers to (a).
  3. What can you check in the operation table to confirm that \(\mathbb{D}\) is closed under matrix multiplication?
  4. Which element of \(\mathbb{D}\) is the identity. We say that matrix is the identity matrix.
  5. Find the (matrix) inverse of each matrix.
  6. Since that matrix multiplication is associative, it follows that \(\mathbb{D}\) is a group. Matrix multiplication is typically not commutative, but is \(\mathbb{D}\text{?}\) Explain.

Subsection 8.3.D Exercises

Subsubsection Exercises for 8.3.A Matrix Products and Powers

Calculate all matrix products and powers by hand, being sure to show the steps of your work. You are welcome to use technology to check your answers.

1. Practice.

Consider the matrices
\begin{equation*} A= \left[ \begin{array}{cc} 1 \amp 2\\ 3 \amp 4 \end{array} \right] B = \left[ \begin{array}{cc} 2 \amp -1\\ 1 \amp 0 \end{array} \right] C= \left[ \begin{array}{cc} 1 \amp 0\\ 0 \amp 0 \end{array} \right] D=\left[ \begin{array}{cc} 0 \amp 0\\ 1 \amp 1 \end{array} \right] I=\left[ \begin{array}{cc} 1 \amp 0\\ 0 \amp 1 \end{array} \right]\text{.} \end{equation*}
Calculate each matrix product: \(AB\text{,}\) \(BA\text{,}\) \(IB\text{,}\) \(AI\text{,}\) \(CD\text{,}\) and \(C^2\text{.}\) Make sure to show work.
Check.
The answers, in some order, are
\begin{equation*} \left[ \begin{array}{rr} 2 \amp -1\\ 1 \amp 0 \end{array} \right] \left[ \begin{array}{rr} -1 \amp 0\\ 1 \amp 2 \end{array} \right] \left[ \begin{array}{cc} 1 \amp 0\\ 0 \amp 0 \end{array} \right] \left[ \begin{array}{rr} 4 \amp -1\\ 10 \amp -3 \end{array} \right] \left[ \begin{array}{cc} 0 \amp 0\\ 0 \amp 0 \end{array} \right] \left[ \begin{array}{cc} 1 \amp 2\\ 3 \amp 4 \end{array} \right] \end{equation*}

2. Practice.

 
  1. For the matrix \(A=\left[ \begin{array}{cc} 1 \amp 0 \\ 1 \amp 1 \end{array} \right]\text{,}\) calculate \(A^2\text{,}\) \(A^3\text{,}\) and \(A^4\text{,}\) and conjecture a formula for \(A^n\) for \(n \ge 1\text{.}\) Make sure to show work.
  2. For the matrix \(B=\left[ \begin{array}{cc} 1 \amp 2 \\ 0 \amp 1 \end{array} \right]\text{,}\) calculate \(B^2\text{,}\) \(B^3\text{,}\) and \(B^4\text{,}\) and conjecture a formula for \(B^n\) for \(n \ge 1\text{.}\) Make sure to show work.
Check.
  1. Hint: \(A^3=\left[ \begin{array}{cc} 1 \amp 0 \\ 3 \amp 1 \end{array} \right]\)
  2. Hint: \(B^3=\left[ \begin{array}{cc} 1 \amp 6 \\ 0 \amp 1 \end{array} \right]\)

3. Practice.

Let \(\mathscr{M}(\mathbb{Z})\) be the set of \(2 \times 2\) matrices with integer entries.
  1. Consider the function \(f: \mathbb{Z} \to \mathscr{M}(\mathbb{Z})\) defined by \(f(n) = \left[ \begin{array}{cc} n \amp 0 \\ 0 \amp 2n \end{array} \right]\text{.}\) Identify the domain and codomain of \(f\) and evaluate \(f(3)\text{.}\)
  2. Consider the function \(\fn{det}: \mathscr{M}(\mathbb{Z}) \to \mathbb{Z}\) defined by \(\fn{det} \left( \left[ \begin{array}{cc} a \amp b \\ c \amp d \end{array} \right] \right) = ad-bc\text{.}\) Identify the domain and codomain of \(\fn{det}\) and evaluate \(\fn{det}\left( \left[ \begin{array}{rr} 2 \amp 0 \\ -4 \amp 3 \end{array} \right] \right)\text{.}\)
  3. Consider the function \(T: \mathscr{M}(\mathbb{Z}) \to \mathscr{M}(\mathbb{Z})\) defined by \(T(A) = AM\) where \(M = \left[ \begin{array}{cr} 1 \amp 2 \\ 0 \amp -1 \end{array} \right]\text{.}\) Identify the domain and codomain of \(T\) and evaluate \(T\left( \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] \right)\text{.}\)
Check.
Hint: Not sure where to start? Look at Example 8.3.4.
  1. Hint: The domain is \(\mathbb{Z}\) and the codomain is \(\mathscr{M}(\mathbb{Z})\text{.}\) Also, \(f(3)\) is a matrix.
  2. Hint: 6
  3. Hint: The answer is not the same as for Example 8.3.4 (c) because they calculated \(MA\) and we want \(AM\text{.}\)

4. Understand.

 
  1. For the unit matrix \(J=\left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 1 \end{array} \right]\text{,}\) calculate \(J^2\text{,}\) \(J^3\text{,}\) and \(J^4\text{,}\) and conjecture a formula for \(J^n\) for \(n \ge 1\text{.}\) Make sure to show work.
  2. For the matrix \(Q=\left[ \begin{array}{cc} 1 \amp 0 \\ 1 \amp 2 \end{array} \right]\text{,}\) calculate \(Q^2\text{,}\) \(Q^3\text{,}\) and \(Q^4\text{,}\) and conjecture a formula for \(Q^n\) for \(n \ge 1\text{.}\) Make sure to show work.
Check.
  1. Hint: All four entries are equal. Careful, they are not \(2^n\text{,}\) but that is close.
  2. Hint: The answer involves 0, 1, \(2^{n-1}\text{,}\) and \(2^n\text{.}\)

5. Understand.

 
  1. For the matrix \(S=\left[ \begin{array}{cc} 0 \amp 1 \\ 0 \amp 0 \end{array} \right]\text{,}\) calculate \(S^2\) and \(S^3\) and explain what you think \(S^{30}\) equals. Make sure to show work.
  2. For the matrix \(W=\left[ \begin{array}{cc} 0 \amp 0 \\ 1 \amp 1 \end{array} \right]\text{,}\) calculate \(W^2\) and \(W^3\) and explain what you think \(W^{30}\) equals. Make sure to show work.
  3. For the matrix \(R=\left[ \begin{array}{cc} 0 \amp -1 \\ 1 \amp 0 \end{array} \right]\text{,}\) calculate \(R^2\text{,}\) \(R^3\text{,}\) and \(R^4\text{,}\) and \(R^5\) and explain what you think \(R^{30}\) equals. Make sure to show work.

6. Understand.

 
  1. Give an example of \(2 \times 2\) matrices \(M\) and \(N\) with \(MN \neq NM\text{.}\)
  2. Give an example of a \(2 \times 2\) matrix \(M\) such that \(M \neq \mathbf{0}\text{,}\) but \(M^2 = \mathbf{0}\text{.}\)
  3. Give an example of a \(2 \times 2\) matrix \(M\) such that \(M \neq I\text{,}\) but \(M^2 = I\text{.}\)

7. Understand.

Let \(\mathscr{M}(\mathbb{Z})\) be the set of \(2 \times 2\) matrices with integer entries.
  1. Consider the function \(f: \mathbb{Z}^+ \to \mathscr{M}(\mathbb{Z})\) defined by \(c(n) = \left[ \begin{array}{cc} 0 \amp 1 \\ -1 \amp 0 \end{array} \right]^n\text{.}\) Evaluate \(c(3)\text{.}\)
  2. Consider the function \(\fn{trace}: \mathscr{M}(\mathbb{Z}) \to \mathbb{Z}\) defined by \(\fn{trace} \left( \left[ \begin{array}{cc} a \amp b \\ c \amp d \end{array} \right] \right) = a+d\text{.}\) Evaluate \(\fn{trace}\left( \left[ \begin{array}{rr} 2 \amp 0 \\ -4 \amp 3 \end{array} \right] \right)\text{.}\)
  3. Consider the function \(\fn{transpose}: \mathscr{M}(\mathbb{Z}) \to \mathscr{M}(\mathbb{Z})\) defined by \(\fn{transpose}\left( \left[ \begin{array}{cc} a \amp b \\ c \amp d \end{array} \right] \right) = \left[ \begin{array}{cc} a \amp c \\ b \amp d \end{array} \right]\text{.}\) Evaluate \(\fn{transpose}\left( \left[ \begin{array}{cc} 1 \amp 2 \\ 3 \amp 5 \end{array} \right] \right)\text{.}\)

8. Understand.

Consider the the function \(\fn{det}\) from Exercise 3 3.(b).
  1. Prove that the function \(\fn{det}\) is onto.
  2. Give a counterexample to prove that the function \(\fn{det}\) is not one-to-one.

9. Understand.

 
  1. Give two examples of elements in the set
    \begin{equation*} \left\{\left[ \begin{array}{cc} 0 \amp a \\ b \amp a+b \end{array} \right] \mid a, b \in \mathbb{Z} \right\} \end{equation*}
  2. Confirm that \(I\) and \(U = \left[ \begin{array}{cc} 0 \amp 0 \\ 0 \amp 1 \end{array} \right]\) are elements of the set \(\mathscr{E} = \left\{ E \in \mathscr{M}(\mathbb{Z}) \mid E^2=E \right\}\)
Check.
Hint: Not sure where to start? Look at Example 8.3.5.
  1. Hint: Your answers should be two different matrices. You can pick any integers you want for \(a\) and \(b\text{.}\)
  2. Hint: You need to check that \(I\) and \(U\) satisfy the implicit condition that defines \(\mathscr{E}\text{.}\)

10. Recap.

Do you know …
  1. How to multiply \(2 \times 2\) matrices?
  2. How to raise a \(2 \times 2\) matrix to a positive integer power?
  3. How to recognize a pattern in the powers of a matrix?

11. Explore.

Give an example of an element in each set and then give an explicit definition of each set.
  1. \(\mathscr{C}= \{ A \in \mathscr{M}(\mathbb{Z}) \mid \fn{trace}(A)=0\}\) where \(\fn{trace}\) is the function defined in Exercise 7 7.(b).
  2. \(\mathscr{D}= \{ A \in \mathscr{M}(\mathbb{Z}) \mid \fn{transpose}(A)=A\}\) where \(\fn{transpose}\) is the function defined in Exercise 7 7.(c).

Subsubsection Exercises for 8.3.B Proof Format: Special Case of Mathematical Induction for Proving an Explicit Formula for the Power of a Matrix

12. Practice.

For the matrix \(A=\left[ \begin{array}{cc} 1 \amp 0 \\ 1 \amp 1 \end{array} \right]\text{,}\) use mathematical induction to prove that \(A^n = \left[ \begin{array}{cc} 1 \amp 0 \\ n \amp 1 \end{array} \right]\) for \(n\ge 1\text{.}\)
Check.
Hint: Copy and edit the proof from Activity 8.3.B.

13. Understand.

For the matrix \(J=\left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 1 \end{array} \right]\text{,}\) use mathematical induction to prove that \(J^n=\left[ \begin{array}{cc} 2^{n-1} \amp 2^{n-1} \\ 2^{n-1} \amp 2^{n-1} \end{array} \right]\) for \(n\ge 1\text{.}\)
Check.
Proof.
Write \(M_n=\left[ \begin{array}{cc} 2^{n-1} \amp 2^{n-1} \\ 2^{n-1} \amp 2^{n-1} \end{array} \right]\text{.}\) Induct on \(n\text{.}\)
First, when \(n=1\) we have \(J^1 = J = \left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 1 \end{array} \right]\) and also \(M_1=\left[ \begin{array}{cc} 2^0 \amp 2^0\\ 2^0 \amp 2^0 \end{array} \right]= \left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 1 \end{array} \right]\text{.}\)
Next, assume \(J^{n-1} = M_{n-1} = \left[ \begin{array}{cc} 2^{n-2} \amp 2^{n-2} \\ 2^{n-2} \amp 2^{n-2} \end{array} \right]\) for some integer \(n \ge 2\text{.}\) Then
\begin{equation*} J^n = JJ^{n-1} =AM_{n-1} = \left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 1 \end{array} \right]\left[ \begin{array}{cc} 2^{n-2} \amp 2^{n-2}\\ 2^{n-2} \amp 2^{n-2} \end{array} \right] = \ldots =M_n \end{equation*}
Fill in the missing algebra using \(2^{n-2}+2^{n-2} = 2 \cdot 2^{n-2} = 2^{n-1}\text{.}\)

14. Understand.

For the matrix \(Q=\left[ \begin{array}{cc} 1 \amp 0 \\ 1 \amp 2 \end{array} \right]\text{,}\) use mathematical induction to prove that \(Q^n=\left[ \begin{array}{cc} 1 \amp 0 \\ 2^{n-1} \amp 2^n \end{array} \right]\) for \(n\ge 1\text{.}\)
Check.
Hint: Copy and edit the proof from the Answers to Exercise 13.

15. Understand.

For the matrix \(M=\left[ \begin{array}{cc} 2 \amp 1 \\ 0 \amp 2 \end{array} \right]\text{,}\) use mathematical induction to prove that \(M^n=\left[ \begin{array}{cc} 2^n \amp n2^{n-1}\\ 0 \amp 2^n \end{array} \right]\) for \(n\ge 1\text{.}\)
Check.
Hint: Copy and edit the proof from the Answers to Exercise 13. Note that
\begin{equation*} 2(n-1)2^{n-2}+1(2^{n-1})=(n-1)2\cdot 2^{n-2} + 2^{n-1}=(n-1)2^{n-1} + 2^{n-1}\text{.} \end{equation*}
To finish the algebra, factor out \(2^{n-1}\) to get
\begin{equation*} ((n-1)+1)2^{n-1} =n2^{n-1} \end{equation*}

16. Understand.

For the matrix \(K=\left[ \begin{array}{cc} 3 \amp 0 \\ 1 \amp 2 \end{array} \right]\text{,}\) use mathematical induction to prove that \(K^n=\left[ \begin{array}{cc} 3^n \amp 0 \\ 3^n-2^n\amp 2^n \end{array} \right]\) for \(n\ge 1\text{.}\)
Check.
Hint: Copy and edit the proof from the Answers to Exercise 13. Note that
\begin{equation*} 1(3^{n-1})+2(3^{n-1}-2^{n-1})=3^{n-1}+2(3^{n-1}) -2(2^{n-1})\text{.} \end{equation*}
To finish the algebra, factor out \(3^{n-1}\) to get
\begin{equation*} (1+2)3^{n-1}-2\cdot 2^{n-1} =3\cdot 3^{n-1}-2\cdot 2^{n-1}=3^n-2^n \end{equation*}

17. Recap.

Do you know …
  1. How use mathematical induction to prove an explicit formula for the power of a matrix?
  2. How to write \(A^n\) in terms of \(A^{n-1}\) for a square matrix \(A\text{?}\)

18. Explore.

Let \(a\) and \(b\) be integers, and consider the diagonal matrix \(D=\left[ \begin{array}{cc} a \amp 0 \\ 0 \amp b \end{array} \right]\text{.}\)
  1. Calculate \(D^2\text{,}\) \(D^3\text{,}\) and \(D^4\text{,}\) and conjecture a formula for \(D^n\) for \(n \ge 1\text{.}\) Your answers will involve \(a\) and \(b\text{.}\) Make sure to show work.
  2. Prove your conjecture using mathematical induction.
Check.
  1. Hint: \(D^n =\left[ \begin{array}{cc} a^n \amp 0 \\ 0 \amp b^n \end{array} \right]\)
  2. Hint: \(M_{n-1}=\left[ \begin{array}{cc} a^{n-1} \amp 0 \\ 0 \amp b^{n-1} \end{array} \right]\)

Subsubsection Exercises for 8.3.C Finite Groups

19. Practice.

In Section 4.3 we saw that the integers mod 5 is the set \(\mathbb{Z}_5 =\{0,1,2,3,4\}\) with addition defined by \(a \oplus b = (a+b) \fn{ mod } 5\text{.}\)
  1. Copy the addition table for \(\mathbb{Z}_5\) in Table 8.3.38 and fill in the missing entries.
    Table 8.3.38. The addition table \(\mathbb{Z}_5\)
    \(\oplus\) 0 1 2 3 4
    0 0 1 2 4
    1 1 2 3 4 0
    2 2 3 0 1
    3 3 4 0 1 2
    4 4 0 1 3
  2. What do you see in the addition table that shows \(\mathbb{Z}_5\) is closed under addition?
  3. What is the additive identity in \(\mathbb{Z}_5\text{?}\)
  4. List each element of \(\mathbb{Z}_5\) and its additive inverse.
  5. Is \(\mathbb{Z}_5\) abelian?
Check.
  1. Hint: The missing entries, in some order, are 2, 3, and 4.
  2. All of the entries are 0, 1, 2, 3, or 4.
  4. Hint: For example, the additive inverse of 2 is 3.
  5. Hint: Check if \(a \oplus b = b \oplus a\) for all \(a, b \in \mathbb{Z}_5\text{.}\)

20. Practice.

Recall that the set of integers mod 10 is \(\mathbb{Z}_{10}=\{0,1,2,3,4,5,6,7,8,9\}\) with multiplication defined by \(a \otimes b = (ab) \fn{ mod } 10\text{.}\) Consider the subset \(\mathbb{U}_{10} = \{1, 3, 7, 9\}\) of \(\mathbb{Z}_{10}\text{.}\)
  1. Copy the multiplication table for \(\mathbb{U}_{10}\) in Table 8.3.40 and fill in the missing entries.
    Table 8.3.40. The multiplication table \(\mathbb{U}_{10}\)
    \(\otimes\) 1 3 7 9
    1 1 3 7 9
    3 3 9 7
    7 7 3
    9 9 7 3
  2. What do you see in the multiplication table that shows \(\mathbb{U}_{10}\) is closed under multiplication?
  3. What is the multiplicative identity in \(\mathbb{U}_{10}\text{?}\)
  4. List each element of \(\mathbb{U}_5\) and its multiplicative inverse.
  5. Is \(\mathbb{U}_{10}\) abelian?
Check.
  1. Hint: The missing entries are 1 or 9.
  2. Hint: What are the entries?
  4. Hint: For example, 9 is its own inverse.
  5. Hint: Check if \(a \otimes b = b \otimes a\) for all \(a, b \in \mathbb{U}_{10}\text{.}\)

21. Understand.

Let \(\mathbb{S}_4\) be the set of all permutations on the set \(\{1,2,3,4\}\) with the operation \(\circ\) which is function composition, introduced in Section 6.4. It turns out that \(\mathbb{S}_4\) is a group.
  1. How many elements are in \(\mathbb{S}_4\text{?}\)
  2. What is the identity element in \(\mathbb{S}_4\text{?}\)
  3. Find the inverse of the permutation \(\left( \begin{array}{cccc} 1 \amp 2 \amp 3 \amp 4 \\ 3 \amp 4 \amp 2 \amp 1 \end{array} \right)\text{.}\)
  4. Give a counterexample to show that \(\mathbb{S}_4\) is not abelian.
Check.
  1. Hint: We counted this number in Example 6.4.4.
  2. Hint: It is the identity permutation as in Example 6.4.4.
  3. Hint: Look at Example 6.4.16.
  4. Hint: Look at Example 6.4.8.

22. Understand.

Consider the following matrices:
\begin{equation*} I = \left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 1 \end{array} \right] R = \left[ \begin{array}{rr} 0 \amp -1 \\ 1 \amp 0 \end{array} \right] S = \left[ \begin{array}{rr} -1 \amp 0 \\ 0 \amp -1 \end{array} \right] T = \left[ \begin{array}{rr} 0 \amp 1 \\ -1 \amp 0 \end{array} \right] \end{equation*}
Let \(\mathbb{Y} = \{I, R, S, T\}\text{.}\)
  1. Write the operation table for \(\mathbb{Y}\text{.}\)
  2. Check that \(\mathbb{Y}\) is a group under function composition.
Check.
  1. Hint: These are the same matrices as in Activity 8.3.C. The operation table should be \(4 \times 4\text{.}\)
  2. Hint: Check each part of the definition of group, as we did in Example 8.3.14.

23. Recap.

Do you know …
  1. What are some examples of groups?
  2. What is the definition of a group?
  3. How can we use the operation table of a group to check the group is closed under the operation?
  4. What the identity of a group is?
  5. How to find the inverse of an element in a group from the operation table?

24. Explore.

The operation table for a finite group is an example of a Latin square which is an \(n \times n\) matrix whose entries are \(n\) different symbols such that each symbol occurs exactly once in each row and each column. Latin squares have application in the design of agricultural experiments in statistics and any Suduko board is a Latin Square.
  1. Construct a \(3 \times 3\) Latin square using the symbols \(1, 2, 3\) where the first row is \(1, 2, 3\text{.}\)
  2. Construct a \(4 \times 4\) Latin square using the symbols \(0, 1, 2, 3\) where the first row is \(0, 1, 2, 3\text{.}\) Hint: Example 8.3.9.
  3. Construct a different \(4 \times 4\) Latin square using the symbols \(0, 1, 2, 3\) where the first row is \(0, 1, 2, 3\text{.}\) Hint: Exercise 20.
  4. Read the Wikipedia entry on Latin Squares. How many different \(5\times 5\) Latin squares are there (on \(1,2,3,4,5\))? Note: Look at the second column in the table.
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