| Date | May 2010 | Marks available | 2 | Reference code | 10M.1.hl.TZ0.2 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | State | Question number | 2 | Adapted from | N/A |
Question
Let S be the set of matrices given by
\(\left[ \begin{array}{l}
a\\
c
\end{array} \right.\left. \begin{array}{l}
b\\
d
\end{array} \right]\) ; \(a,b,c,d \in \mathbb{R}\), \(ad - bc = 1\)
The relation \(R\) is defined on \(S\) as follows. Given \(\boldsymbol{A}\) , \(\boldsymbol{B} \in S\) , \(\boldsymbol{ARB}\) if and only if there exists \(\boldsymbol{X} \in S\) such that \(\boldsymbol{A} = \boldsymbol{BX}\) .
Show that \(R\) is an equivalence relation.
The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State, with a reason, whether or not there are any non-zero values of \(n\) , other than \(1\), for which \(R\) is an equivalence relation.
Markscheme
since \(\boldsymbol{A} = \boldsymbol{AI}\) where \(\boldsymbol{I}\) is the identity A1
and \(\det (\boldsymbol{I}) = 1\) , A1
\(R\) is reflexive
\(\boldsymbol{ARB} \Rightarrow \boldsymbol{A} = \boldsymbol{BX}\) where \(\det (\boldsymbol{X}) = 1\) M1
it follows that \(\boldsymbol{B} = \boldsymbol{A}{\boldsymbol{X}^{ - 1}}\) A1
and \(\det ({\boldsymbol{X}^{ - 1}}) = \det{(\boldsymbol{X})^{ - 1}} = 1\) A1
\(R\) is symmetric
\(\boldsymbol{ARB}\) and \(\boldsymbol{BRC} \Rightarrow \boldsymbol{A} = \boldsymbol{BX}\) and \(\boldsymbol{B} = \boldsymbol{CY}\) where \(\det (\boldsymbol{X}) = \det (\boldsymbol{Y}) = 1\) M1
it follows that \(\boldsymbol{A} = \boldsymbol{CYX}\) A1
\(\det (\boldsymbol{YX}) = \det (\boldsymbol{Y})\det (\boldsymbol{X}) = 1\) A1
\(R\) is transitive
hence \(R\) is an equivalence relation AG
[8 marks]
for reflexivity, we require \(\boldsymbol{ARA}\) so that \(\boldsymbol{A} = \boldsymbol{AI}\) (for all \(\boldsymbol{A} \in S\) ) M1
since \(\det (\boldsymbol{I}) = 1\) and we require \(\boldsymbol{I} \in S\) the only possibility is \(n = 1\) A1
[2 marks]
Examiners report
This question was not well done in general, again illustrating that questions involving both matrices and equivalence relations tend to cause problems for candidates. A common error was to assume, incorrectly, that ARB and BRC \( \Rightarrow A = BX\) and \(B = CX\) , not realizing that a different "\(x\)" is required each time. In proving that \(R\) is an equivalence relation, consideration of the determinant is necessary in this question although many candidates neglected to do this.
In proving that \(R\) is an equivalence relation, consideration of the determinant is necessary in this question although many candidates neglected to do this.
