4.03b Matrix operations: addition, multiplication, scalar

4. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & - 2 & - 7 \\ 3 & k & 2 \\ 1 & 1 & 4 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { c c c } 4 k - 2 & 1 & 7 k - 4 \\ - 10 & 3 & - 19 \\ 3 - k & - 1 & 6 - k \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of the constant \(c\) for which $$\mathbf { A B } = ( 3 k + c ) \mathbf { I }$$
2. Hence determine the value of \(k\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Given that \(\mathbf { A } ^ { - 1 }\) does exist,
3. write down \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
4. Use the answer to part (c) to solve the simultaneous equations $$\begin{aligned} - x - 2 y - 7 z & = 10 \\ 3 x + k y + 2 z & = 3 \\ x + y + 4 z & = 1 \end{aligned}$$ giving the values of \(x , y\) and \(z\) in simplest form in terms of \(k\).

1. Prove by induction that for \(n \in \mathbb { N }\)

$$\left( \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & - 2 n \\ 0 & 1 \end{array} \right)$$

3. $$\mathbf { M } = \left( \begin{array} { r r } - 2 & 5 \\ 6 & k \end{array} \right)$$ where \(k\) is a constant.
Given that $$\mathbf { M } ^ { 2 } + 11 \mathbf { M } = a \mathbf { I }$$ where \(a\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,

1. 1. determine the value of \(a\)
   2. show that \(k = - 9\)
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf { M }\).
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer.

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).

1. Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
3. A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).

6 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { l l } 1 & 2 \\ 1 & a \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & b \end{array} \right)\).

1. Find the matrix \(\mathbf { A B }\).
2. State the conditions on \(a\) and \(b\) for \(\mathbf { A B }\) to be a singular matrix. \(P Q R S\) is a quadrilateral and the vertices \(P , Q , R\) and \(S\) are in clockwise order. A transformation, T , is represented by the matrix \(\mathbf { A B }\).
3. State the effect on both the area and also the orientation of the image of \(P Q R S\) under T in each of the following cases.
   1. \(\quad a = 1\) and \(b = 1\)
   2. \(\quad a = 2\) and \(b = 3\)

5 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1 \\ - 8 & 4 & 0 \\ 19 & - 8 & - 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Hence write down \(\mathbf { A } ^ { - 1 }\).
3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0 \\ 2 x + 5 y + 2 z = 1 \\ 3 x - 2 y - z = 4 \end{array}$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
   2. Find this unique solution.

6 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3 \\ 3 & - \sqrt { 3 } \end{array} \right]$$

1. 1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Show that the matrix \(\mathbf { M }\) can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ } \\ \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where \(q\) is a real number. Give the value of \(q\) in surd form.
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection. Find:
   1. the scale factor of the enlargement;
   2. the equation of the mirror line of the reflection.
3. Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).

4 It is given that $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 4 \\ 3 & 1 \end{array} \right]$$ and that \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Show that \(( \mathbf { A } - \mathbf { I } ) ^ { 2 } = k \mathbf { I }\) for some integer \(k\).
2. Given further that $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 3 \\ p & 1 \end{array} \right]$$ find the integer \(p\) such that $$( \mathbf { A } - \mathbf { B } ) ^ { 2 } = ( \mathbf { A } - \mathbf { I } ) ^ { 2 }$$

6 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a rectangle \(R _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}

1. The rectangle \(R _ { 1 }\) is mapped onto a second rectangle, \(R _ { 2 }\), by a transformation with matrix \(\left[ \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right]\).
   1. Calculate the coordinates of the vertices of the rectangle \(R _ { 2 }\).
   2. On Figure 1, draw the rectangle \(R _ { 2 }\).
2. The rectangle \(R _ { 2 }\) is rotated through \(90 ^ { \circ }\) clockwise about the origin to give a third rectangle, \(R _ { 3 }\).
   1. On Figure 1, draw the rectangle \(R _ { 3 }\).
   2. Write down the matrix of the rotation which maps \(R _ { 2 }\) onto \(R _ { 3 }\).
3. Find the matrix of the transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { l l } 3 & 4 \\ 4 & 3 \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 0 & 2 \\ 2 & 0 \end{array} \right]$$

1. Calculate the matrices:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A B }\).
2. Show that \(\mathbf { A } + \mathbf { B } - \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   (2 marks)

5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$

1. Find the matrix:
   1. \(\mathbf { M } ^ { 2 }\);
   2. \(\mathbf { M } ^ { 4 }\).
2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
3. Find the matrix \(\mathbf { M } ^ { 2006 }\).

4 M is the set of all \(2 \times 2\) matrices \(\mathrm { m } ( a , b )\) where \(a\) and \(b\) are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b \\ 0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$

1. Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
2. Determine whether the group is commutative. The set \(\mathrm { N } _ { k }\) consists of all \(2 \times 2\) matrices \(\mathrm { m } ( k , b )\) where \(k\) is a fixed positive integer and \(b\) can take any integer value.
3. Prove that \(\mathrm { N } _ { k }\) is closed under matrix multiplication if and only if \(k = 1\). Now consider the set P consisting of the matrices \(\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )\) and \(\mathrm { m } ( 1,3 )\). The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
4. Show that \(( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )\).
5. Construct the group combination table for P . The group R consists of the set \(\{ e , a , b , c \}\) combined under the operation *. The identity element is \(e\), and elements \(a , b\) and \(c\) are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
6. Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}

10

1. Show that \(\operatorname { det } \mathbf { A } = a + \mathrm { i }\) where \(a\) is an integer to be determined. 10 Matrix A is given by 10
2. Matrix B is given by $$\mathbf { B } = \left[ \begin{array} { c c } 14 - 2 \mathrm { i } & b \\ c & d \end{array} \right] \quad \text { and } \quad \mathbf { A B } = p$$ where \(b , c , d \in \mathbb { C }\) and \(p \in \mathbb { N }\) Find \(b , c , d\) and \(p\)

11 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } 3 \mathrm { i } & - 2 \\ a & - \mathrm { i } \end{array} \right] \quad \text { and } \quad \mathbf { B } = \left[ \begin{array} { c c } 4 & 5 \\ - 2 \mathrm { i } & - 1 \end{array} \right]$$ where \(a\) is a real number. Calculate the product \(\mathbf { A B }\) in terms of \(a\) Give your answer in its simplest form.
[0pt] [3 marks]

13

1. The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant. Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m \\ 2 m & m ^ { 2 } - 1 \end{array} \right]\) You may use the result in the formulae booklet. 13
2. \(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right]\) Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
   13
3. (i) The diagram below shows a point \(P\) and the line \(y = m x\) Draw four lines on the diagram to demonstrate the result proved in part (b).
   Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13 (c) (ii) Explain how your completed diagram shows the result proved in part (b).
   13
4. The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 } \\ \frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\) Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\) Fully justify your answer.
   [0pt] [4 marks]

2 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.

1. State, with a reason, which one of A and B is true.
2. Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).

9 Matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b \\ 0 & 1 & 0 \\ b & 0 & a \end{array} \right)\) where \(a\) and \(b\) are constants.

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). The constants \(a\) and \(b\) are given by \(a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )\) and \(b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )\).
2. By determining exact expressions for \(a b\) and \(a ^ { 2 } - b ^ { 2 }\) and using the result from part (a), show that \(\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1 \\ 0 & 2 & 0 \\ 1 & 0 & \sqrt { 3 } \end{array} \right)\) where \(k\) is a real number whose value is to be determined.
3. Find \(\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }\) and \(\mathbf { R } ^ { 24 }\).
4. Describe fully the transformation represented by \(\mathbf { R }\). \section*{END OF QUESTION PAPER}

5 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } - 1 & 0 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).

1. Use \(\mathbf { A }\) and \(\mathbf { B }\) to disprove the proposition: "Matrix multiplication is commutative". Matrix \(\mathbf { B }\) represents the transformation \(\mathrm { T } _ { \mathrm { B } }\).
2. Describe the transformation \(\mathrm { T } _ { \mathrm { B } }\).
3. By considering the inverse transformation of \(\mathrm { T } _ { \mathrm { B } }\), determine \(\mathbf { B } ^ { - 1 }\). Matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 3 \end{array} \right)\) and represents the transformation \(\mathrm { T } _ { \mathrm { C } }\).
   The transformation \(\mathrm { T } _ { \mathrm { BC } }\) is transformation \(\mathrm { T } _ { \mathrm { C } }\) followed by transformation \(\mathrm { T } _ { \mathrm { B } }\).
   An object shape of area 5 is transformed by \(\mathrm { T } _ { \mathrm { BC } }\) to an image shape \(N\).
4. Determine the area of \(N\).

4 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\).

1 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.

1. State, with a reason, which one of A and B is true.
2. Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).

1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2 \\ 9 \\ 0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0 \\ - 8 \\ 3 \end{array} \right) .$$

1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).

9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
2. Explain how your answer to part (i) relates to this information.
3. By investigating the invariant points of the reflection, find the equation of the mirror line.
4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS
   4755
   \textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}

   Tuesday 7 JUNE 2005	Afternoon	1 hour 30 minutes
   Additional materials: Answer booklet Graph paper MEI Examination Formulae and Tables (MF2)

   TIME 1 hour 30 minutes
   • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
   • Answer all the questions.
   • You are permitted to use a graphical calculator in this paper.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • You are advised that an answer may receive no marks unless you show sufficient defail of the working to indicate that a correct method is being used.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The total number of marks for this paper is 72.

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.