4.03b Matrix operations: addition, multiplication, scalar

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2 \\ 0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2 \\ 0 & 1 \end{array} \right) .$$

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 ( \(\mathbf { v }\) ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?

1 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & - 3 \\ 1 & 4 \end{array} \right) , \mathbf { C } = \left( \begin{array} { r r } 1 & - 1 \\ 0 & 2 \\ 0 & 1 \end{array} \right)\).

1. Calculate, where possible, \(2 \mathbf { B } , \mathbf { A } + \mathbf { C } , \mathbf { C A }\) and \(\mathbf { A } - \mathbf { B }\).
2. Show that matrix multiplication is not commutative.

9 Matrices \(\mathbf { M }\) and \(\mathbf { N }\) are given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 2 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { r r } 1 & - 3 \\ 1 & 4 \end{array} \right)\).

1. Find \(\mathbf { M } ^ { - 1 }\) and \(\mathbf { N } ^ { - 1 }\).
2. Find \(\mathbf { M N }\) and \(( \mathbf { M N } ) ^ { - \mathbf { 1 } }\). Verify that \(( \mathbf { M N } ) ^ { - 1 } = \mathbf { N } ^ { - 1 } \mathbf { M } ^ { - 1 }\).
3. The result \(( \mathbf { P Q } ) ^ { - 1 } = \mathbf { Q } ^ { - 1 } \mathbf { P } ^ { - 1 }\) is true for any two \(2 \times 2\), non-singular matrices \(\mathbf { P }\) and \(\mathbf { Q }\). The first two lines of a proof of this general result are given below. Beginning with these two lines, complete the general proof. $$\begin{aligned} & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q } = \mathbf { I } \\ \Rightarrow & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q Q } \mathbf { Q } ^ { - 1 } = \mathbf { I Q } ^ { - 1 } \end{aligned}$$

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

1. Find BA.
2. A plane shape of area 3 square units is transformed using matrix \(\mathbf { A }\). The image is transformed using matrix B. What is the area of the resulting shape?

5 You are given that \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 4 \\ 3 & 2 & 5 \\ 4 & 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 1 & 0 & 2 \\ 14 & - 14 & 7 \\ - 5 & 7 & - 4 \end{array} \right)\).

1. Calculate AB.
2. Write down \(\mathbf { A } ^ { - 1 }\).

5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities \(0.7,0.1,0.2\) respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities \(0.1,0.8,0.1\) respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities \(0.3,0.6,0.1\) respectively. The situation is modelled as a Markov chain with four states.

1. Write down the transition matrix.
2. Find the probabilities that level 14 is set in each location.
3. Find the probability that level 15 is set in the same location as level 14 .
4. Find the level at which the probability of being set in Chiworld first exceeds 0.5.
5. Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities \(0.9,0.1\) respectively.
6. By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
7. Find the new transition probabilities after a level set in Chiworld. }{www.ocr.org.uk}) after the live examination series.
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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.

8 The set \(M\) of matrices \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), where \(a , b , c\) and \(d\) are real and \(a d - b c = 1\), forms a group \(( M , \times )\) under matrix multiplication. \(R\) denotes the set of all matrices \(\left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)\).

1. Prove that ( \(R , \times\) ) is a subgroup of ( \(M , \times\) ).
2. By considering geometrical transformations in the \(x - y\) plane, find a subgroup of \(( R , \times )\) of order 6 . Give the elements of this subgroup in exact numerical form. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ a & 5 \end{array} \right)\). Find

1. \(\mathbf { A } ^ { - 1 }\),
2. \(2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right)\).

6

1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 2 \\ 3 & 4 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Find A-4I.
2. Given that \(\mathbf { A }\) is singular, find the value of \(a\).

5

1. The transformation T is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\). Give a geometrical description of T .
2. The transformation T is equivalent to a reflection in the line \(y = - x\) followed by another transformation S . Give a geometrical description of S and find the matrix that represents S .

10 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).

1. Find \(\mathbf { M } ^ { 2 }\) and \(\mathbf { M } ^ { 3 }\).
2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.
4. Describe fully the single geometrical transformation represented by \(\mathbf { M } ^ { 10 }\).

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

1. \(2 \mathbf { A } + \mathbf { B }\),
2. \(\mathbf { A C }\),
3. CB.

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 4 & 6 \\ 3 & - 5 \end{array} \right)\), and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Given that \(p \mathbf { A } + q \mathbf { B } = \mathbf { I }\), find the values of the constants \(p\) and \(q\).

7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 2 & 1 \end{array} \right)\).

1. Show that \(\mathbf { M } ^ { 4 } = \left( \begin{array} { l l } 81 & 0 \\ 80 & 1 \end{array} \right)\).
2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\), where \(n\) is a positive integer.
3. Use induction to prove that your answer to part (ii) is correct.

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 5 & 0 \\ 0 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find the values of the constants \(a\) and \(b\) for which \(a \mathbf { A } + b \mathbf { B } = \mathbf { I }\).

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 2 & a \\ 4 & 1 \end{array} \right)\). I denotes the \(2 \times 2\) identity matrix. Find

1. \(\mathbf { A } + 3 \mathbf { B } - 4 \mathbf { I }\),
2. AB.

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

1. \(\mathbf { A B }\),
2. \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\).

9

1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { X }\).
2. The matrix \(\mathbf { Z }\) is given by \(\mathbf { Z } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } ( 2 + \sqrt { 3 } ) \\ - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } ( 1 - 2 \sqrt { 3 } ) \end{array} \right)\). The transformation represented by \(\mathbf { Z }\) is equivalent to the transformation represented by \(\mathbf { X }\), followed by another transformation represented by the matrix \(\mathbf { Y }\). Find \(\mathbf { Y }\).
3. Describe fully the geometrical transformation represented by \(\mathbf { Y }\).

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } 2 & 1 \\ - 4 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find

1. \(4 \mathbf { A } - \mathbf { B } + 2 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\),
3. \(\left( \mathbf { A B } ^ { - 1 } \right) ^ { - 1 }\).

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l l } b & 0 & 5 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r } 6 \\ 4 \\ - 1 \end{array} \right)\). Find

1. \(5 \mathbf { A } - 3 \mathbf { B }\),
2. BC,
3. CA .

3 Fig. 3 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h] \end{figure}

1. Write down the matrix \(\mathbf { P }\) representing this transformation.
2. The parallelogram \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is transformed by the matrix \(\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\). Find the coordinates of the vertices of its image, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\), following this transformation.
3. Describe fully the transformation represented by \(\mathbf { Q P }\).

10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 3 & 4 & - 1 \\ 1 & - 1 & k \\ - 2 & 7 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r c } 11 & - 5 & - 7 \\ 1 & 11 & 5 + k \\ - 5 & 29 & 7 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } 42 & \alpha & 4 k - 8 \\ 10 - 5 k & - 16 + 29 k & - 12 + 6 k \\ 0 & 0 & \beta \end{array} \right)\).

1. Show that \(\alpha = 0\) and \(\beta = 28 + 7 k\).
2. Find \(\mathbf { A B }\) when \(k = 2\).
3. For the case when \(k = 2\) write down the matrix \(\mathbf { A } ^ { - 1 }\).
4. Use the result from part (iii) to solve the following simultaneous equations. $$\begin{aligned} 3 x + 4 y - z & = 1 \\ x - y + 2 z & = - 9 \\ - 2 x + 7 y - 3 z & = 26 \end{aligned}$$