4.03a Matrix language: terminology and notation

5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & - \frac { 1 } { 2 } \sqrt { 2 } \\ \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right) \left( \begin{array} { c c } 1 & k \\ 0 & 1 \end{array} \right)\), where \(k\) is a constant.

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
2. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find, in terms of \(k\), the single matrix which transforms triangle \(D E F\) onto triangle \(A B C\).
3. Find the set of values of \(k\) for which the transformation represented by \(\mathbf { M }\) has no invariant lines through the origin.

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 1 & 0 \\ 0 & k \end{array} \right) \left( \begin{array} { c c } 1 & 0 \\ k & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) or 1 .

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(\mathrm { y } = \frac { \mathrm { k } ^ { 2 } } { 1 - \mathrm { k } } \mathrm { x }\). [4]
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find the value of \(k\) for which the area of triangle \(D E F\) is equal to the area of triangle \(A B C\).

5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.

1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).
3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
   Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
   The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
2. Find, in terms of \(k\), the area of parallelogram \(O P Q R\) and the matrix which transforms \(O P Q R\) onto the unit square.
3. Show that the line through the origin with gradient \(\frac { 1 } { k - 1 }\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$

1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-08_2715_31_106_2016} Let \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).
3. Give full details of the transformation represented by \(\mathbf { M }\).
4. Find the matrix \(\mathbf { N }\) such that \(\mathbf { N M } = \mathbf { C A B }\).

8

1. Find the value of \(a\) for which the system of equations $$\begin{gathered} 3 x + a y = 0 \\ 5 x - y = 0 \\ x + 3 y + 2 z = 0 \end{gathered}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 0 & 0 \\ 5 & - 1 & 0 \\ 1 & 3 & 2 \end{array} \right)$$
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
3. 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.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 9 & 5 \\ 5 & - 8 & 5 \\ 1 & - 1 & 2 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }\), where \(p\) and \(q\) are constants to be determined.

5 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 18 & 5 & - 11 \\ 8 & 6 & - 4 \\ 32 & 10 & - 20 \end{array} \right)$$

1. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 20 \lambda + 48 = 0\) and hence find the eigenvalues of \(\mathbf { A }\).
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).

7 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 4 & 2 \\ 0 & - 1 & 1 \\ 0 & 0 & 2 \end{array} \right) .$$

1. State the eigenvalues of \(\mathbf { P }\).
2. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
   The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues \(b , - 1,1\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right)$$ respectively.
3. Find \(\mathbf { A }\) in terms of b.

7. Given that \(\mathbf { X } = \left( \begin{array} { c c } 2 & a \\ - 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),

1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

1. \(\mathbf { M } = \left( \begin{array} { r r r } 0 & 1 & 9 \\ 1 & 4 & k \\ 1 & 0 & - 3 \end{array} \right)\), where \(k\) is a constant.

Given that \(\left( \begin{array} { r } 7 \\ 19 \\ 1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { r } 7 \\ 19 \\ 1 \end{array} \right)\),
2. show that \(k = - 7\)
3. find the other two eigenvalues of the matrix \(\mathbf { M }\). The image of the vector \(\left( \begin{array} { c } p \\ q \\ r \end{array} \right)\) under the transformation represented by \(\mathbf { M }\) is \(\left( \begin{array} { r } - 6 \\ 21 \\ 5 \end{array} \right)\).
4. Find the values of the constants \(p , q\) and \(r\).

2. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 3 & a \\ 2 & 0 & 1 \\ 1 & - 2 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 4 \\ 3 & - 2 & 3 \\ 1 & 2 & b \end{array} \right)$$ where \(a\) and \(b\) are constants.

1. Write down \(\mathbf { A } ^ { \mathrm { T } }\) in terms of \(a\).
2. Calculate \(\mathbf { A B }\), giving your answer in terms of \(a\) and \(b\).
3. Hence show that $$( \mathbf { A B } ) ^ { \mathrm { T } } = \mathbf { B } ^ { \mathrm { T } } \mathbf { A } ^ { \mathrm { T } }$$

4. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right)$$

1. Show that 6 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
2. Find a normalised eigenvector corresponding to the eigenvalue 6

2. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 6 \end{array} \right)$$

1. Find the eigenvalues and corresponding normalised eigenvectors of the matrix \(\mathbf { A }\).
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).

7. \(\quad \mathbf { A } = \left( \begin{array} { c c c } 2 & \mathrm { k } & 0 \\ 1 & 1 & 0 \\ 0 & - 2 & 1 \end{array} \right)\), where k is a constant. Given that \(\left( \begin{array} { c } 9 \\ 3 \\ - 2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),

1. show that \(\mathrm { k } = 6\),
2. find the eigenvalues of \(\mathbf { A }\). A transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { A }\).
   The point P has coordinates \(( \mathrm { t } - 2 , \mathrm { t } , 2 \mathrm { t } )\) where t is a parameter.
3. Show that, for any value of \(t\), the transformation \(T\) maps \(P\) onto a point on the line with equation \(x - 4 y - 4 = 0\) (5)

6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\),
2. show that \(k = 3\),
3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
   The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\).
4. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For the eigenvalue 4, find a corresponding eigenvector. The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\). The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
3. Find a vector equation for the line \(l _ { 2 }\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 2 \\ - 2 \end{array} \right) ,$$ where \(s\) and \(t\) are real parameters. The plane \(\Pi _ { 1 }\) is transformed to the plane \(\Pi _ { 2 }\) by the transformation represented by the matrix \(\mathbf { T }\), where $$\mathbf { T } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & 2 & - 1 \\ 0 & 1 & 2 \end{array} \right)$$ Find an equation of the plane \(\Pi _ { 2 }\) in the form r.n=p

6. It is given that \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l l } 4 & 2 & 3 \\ 2 & b & 0 \\ a & 1 & 8 \end{array} \right)$$ and \(a\) and \(b\) are constants.

1. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\).
2. Find the values of \(a\) and \(b\).
3. Find the other eigenvalues of \(\mathbf { A }\).

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 1 & a \\ 2 & b & c \\ - 1 & 0 & 1 \end{array} \right) , \text { where } a , b \text { and } c \text { are constants. }$$

1. Given that \(\mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - \mathbf { k }\) are two of the eigenvectors of \(\mathbf { M }\), find
   1. the values of \(a , b\) and \(c\),
   2. the eigenvalues which correspond to the two given eigenvectors.
2. The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 1 & 0 \\ 2 & 1 & d \\ - 1 & 0 & 1 \end{array} \right) \text {, where } d \text { is constant, } d \neq - 1$$ Find
   1. the determinant of \(\mathbf { P }\) in terms of \(d\),
   2. the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(d\).

6. The symmetric matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) with eigenvalues 5, 2 and - 1 respectively.

1. Find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$ Given that \(\mathbf { P } ^ { - 1 } = \mathbf { P } ^ { \mathrm { T } }\)
2. show that $$\mathbf { M } = \mathbf { P D P } ^ { - 1 }$$
3. Hence find the matrix \(\mathbf { M }\).

3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\),

1. find the value of \(k\).
2. Hence find the other two eigenvalues of \(\mathbf { M }\).
3. Find an eigenvector corresponding to the eigenvalue 3
   3. \(\quad \mathbf { M } = \left( \begin{array} { r c c } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\), (a) find the value of \(k\).

5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division. This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 . The following transition probabilities apply to this team from 2007 onwards.

• When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
• When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
• When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .

This process is modelled as a Markov chain with three states corresponding to the three divisions.

1. Write down the transition matrix.
2. Determine in which division the team is most likely to be playing in 2014.
3. Find the equilibrium probabilities for each division for this team. In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.
   -The transition probabilities from Divisions 1 and 2 remain the same as before.
   • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
   The team plays in Division 2 in 2015.
   The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.
4. Write down the transition matrix which applies from 2015.
5. Find the probability that the team is still playing in the league in 2020.
6. Find the first year for which the probability that the team is out of the league is greater than 0.5 .

5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes \(A , B\) and \(C\). The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route \(A\) was used on the previous day, route \(A\), \(B\) or \(C\) will be used, with probabilities \(0.1,0.4,0.5\) respectively.
If route \(B\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.7,0.2,0.1\) respectively.
If route \(C\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.1,0.6,0.3\) respectively. The situation is modelled as a Markov chain with three states.

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probability that route \(B\) is used on the 7th day.
3. Find the probability that the same route is used on the 7th and 8th days.
4. Find the probability that the route used on the 10th day is the same as that used on the 7th day.
5. Given that \(\mathbf { P } ^ { n } \rightarrow \mathbf { Q }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { Q }\) (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix \(\mathbf { Q }\). The computer program is now to be changed, so that the long-run probabilities for routes \(A , B\) and \(C\) will become \(0.4,0.2\) and 0.4 respectively. The transition probabilities after routes \(A\) and \(B\) remain the same as before.
6. Find the new transition probabilities after route \(C\).
7. A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day.

5 In this question, give probabilities correct to 4 decimal places.
A contestant in a game-show starts with one, two or three 'lives', and then performs a series of tasks. After each task, the number of lives either decreases by one, or remains the same, or increases by one. The game ends when the number of lives becomes either four or zero. If the number of lives is four, the contestant wins a prize; if the number of lives is zero, the contestant loses and leaves with nothing. At the start, the number of lives is decided at random, so that the contestant is equally likely to start with one, two or three lives. The tasks do not involve any skill, and after every task:

• the probability that the number of lives decreases by one is 0.5 ,
• the probability that the number of lives remains the same is 0.05 ,
• the probability that the number of lives increases by one is 0.45 .

This is modelled as a Markov chain with five states corresponding to the possible numbers of lives. The states corresponding to zero lives and four lives are absorbing states.

1. Write down the transition matrix \(\mathbf { P }\).
2. Show that, after 8 tasks, the probability that the contestant has three lives is 0.0207 , correct to 4 decimal places.
3. Find the probability that, after 10 tasks, the game has not yet ended.
4. Find the probability that the game ends after exactly 10 tasks.
5. Find the smallest value of \(N\) for which the probability that the game has not yet ended after \(N\) tasks is less than 0.01 .
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
7. Find the probability that the contestant wins a prize. The beginning of the game is now changed, so that the probabilities of starting with one, two or three lives can be adjusted.
8. State the maximum possible probability that the contestant wins a prize, and how this can be achieved.
9. Given that the probability of starting with one life is 0.1 , and the probability of winning a prize is 0.6 , find the probabilities of starting with two lives and starting with three lives. }{www.ocr.org.uk}) after the live examination series.
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