4.03a Matrix language: terminology and notation

5 In this question, give probabilities correct to 4 decimal places.
The speeds of vehicles are measured on a busy stretch of road and are categorised as A (not more than 30 mph ), B (more than 30 mph but not more than 40 mph ) or C (more than 40 mph ).

• Following a vehicle in category A , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.9,0.07,0.03\) respectively.
• Following a vehicle in category B , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.3,0.6,0.1\) respectively.
• Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.

This is modelled as a Markov chain with three states corresponding to the categories A, B, C. The speed of the first vehicle is measured as 28 mph .

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probabilities that the 10th vehicle is in each of the three categories.
3. Find the probability that the 12th and 13th vehicles are in the same category.
4. Find the smallest value of \(n\) for which the probability that the \(n\)th and \(( n + 1 )\) th vehicles are in the same category is less than 0.8, and give the value of this probability.
5. Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity, and hence write down the equilibrium probabilities for the three categories.
7. Find the probability that, after many vehicles have passed by, the next three vehicles are all in category A. On a new stretch of road, the same categories are used but some of the transition probabilities are different.
   • Following a vehicle in category A , the probability that the next vehicle is in category B is equal to the probability that it is in category C .
   • Following a vehicle in category B , the probability that the next vehicle is in category A is equal to the probability that it is in category C .
   • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
   In the long run, the proportions of vehicles in categories A, B, C are 50\%, 40\%, 10\% respectively.
8. Find the transition matrix for the new stretch of road.

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 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point \(( 10,50 )\) under transformation T .
2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
6. Find the \(2 \times 2\) matrix which represents the transformation.
7. Show that this matrix is singular. Relate this result to the transformation.

3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h] \end{figure}

1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = x\) and has the same \(x\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point ( \(- 3,7\) ) under transformation T .
2. Write down the image of the point \(( x , y )\) under transformation T .
3. Find the \(2 \times 2\) matrix which represents the transformation.
4. Describe the transformation M represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
5. Find the matrix representing the composite transformation of T followed by M .
6. Find the image of the point \(( x , y )\) under this composite transformation. State the equation of the line on which all of these images lie.

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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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 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 \(a \mathbf { A } + b \mathbf { B } = \mathbf { I }\).

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 \(\mathbf { Q P }\).

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

1. Calculate, where they exist, \(\mathbf { A B } , \mathbf { C A } , \mathbf { B } + \mathbf { D }\) and \(\mathbf { A C }\) and indicate any that do not exist.
2. Matrices \(\mathbf { B }\) and \(\mathbf { D }\) represent transformations B and D respectively. Find the single matrix that represents transformation B followed by transformation D.

9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.

2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 2 & - 5 \\ 3 & 7 \end{array} \right)\). \(\mathbf { M } \binom { x } { y } = \binom { 9 } { - 1 }\) represents two simultaneous equations.

1. Write down these two equations.
2. Find \(\mathbf { M } ^ { - 1 }\) and use it to solve the equations.

9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.

1. Describe fully the transformations P and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
   Triangle T is transformed first by transformation P , then by transformation Q .
2. Find the single matrix that represents this composite transformation.
3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.

1

1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix for a reflection in the line \(y = x\).
3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
4. What single transformation is equivalent to this composite transformation?

9 The simultaneous equations $$\begin{aligned} & 2 x - y = 1 \\ & 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).

1. Write down the matrix \(\mathbf { M }\).
2. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists. Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
3. What can you say about the solutions of the equations when \(k = - \frac { 3 } { 2 }\) ?
4. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
   (A) when \(k = 5\) and \(b = 21\),
   (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
   (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\). RECOGNISING ACHIEVEMENT

1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation B .

1. Describe the transformations A and B .
2. Find the matrix representing the composite transformation consisting of A followed by B .
3. What single transformation is represented by this matrix?

2 Fig. 2 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 \(\mathbf { T }\) representing this transformation. The quadrilateral \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is reflected in the \(x\)-axis to give a new quadrilateral, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
2. Write down the matrix representing reflection in the \(x\)-axis.
3. Find the single matrix that will transform OABC onto \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).

3 Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector, then \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector. The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 4 & - 6 & - 6 \\ 5 & 11 & 10 \end{array} \right)$$ has \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) as an eigenvector. Find the corresponding eigenvalue. The other two eigenvalues of \(\mathbf { A }\) are 1 and 2, with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. The matrix \(\mathbf { B }\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\), \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
[0pt] [You are not required to evaluate \(\mathbf { P } ^ { - 1 }\).]

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 2 & - 1 & - 1 \\ 1 & 3 & - 1 & 0 \\ 1 & 0 & 3 & 1 \\ 0 & 3 & - 4 & - 1 \end{array} \right) .$$ The range space of T is denoted by \(V\).

1. Determine the dimension of \(V\).
2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 2 \\ 3 \\ 0 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \\ - 4 \end{array} \right)\) are linearly independent.
3. Write down a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
4. State, with a reason, whether \(W\) is a vector space.
5. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(y - z - t \neq 0\).

9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4 \\ 1 & 5 & - 1 \\ 2 & 1 & 5 \end{array} \right)$$ has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\).
[0pt] [The evaluation of \(\mathbf { P } ^ { - 1 }\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k ^ { n } \mathbf { A } ^ { n }\) tends to the zero matrix as \(n \rightarrow \infty\).

The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices \(\mathbf { M } _ { 1 }\) and \(\mathbf { M } _ { 2 }\), respectively, where $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 2 \\ 1 & 4 & 7 & 8 \\ 1 & 7 & 11 & 13 \\ 1 & 2 & 5 & 5 \end{array} \right) , \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 2 & 0 & - 1 & - 1 \\ 5 & 1 & - 3 & - 3 \\ 3 & - 1 & - 1 & - 1 \\ 13 & - 1 & - 6 & - 6 \end{array} \right) .$$

1. Find a basis for \(R _ { 1 }\), the range space of \(\mathrm { T } _ { 1 }\).
2. Find a basis for \(K _ { 2 }\), the null space of \(\mathrm { T } _ { 2 }\), and hence show that \(K _ { 2 }\) is a subspace of \(R _ { 1 }\). The set of vectors which belong to \(R _ { 1 }\) but do not belong to \(K _ { 2 }\) is denoted by \(W\).
3. State whether \(W\) is a vector space, justifying your answer. The linear transformation \(\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is the result of applying \(\mathrm { T } _ { 1 }\) and then \(\mathrm { T } _ { 2 }\), in that order.
4. Find the dimension of the null space of \(\mathrm { T } _ { 3 }\).

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 1 & - 1 \\ - 4 & - 1 & 4 \\ 0 & - 1 & 5 \end{array} \right)$$ Given that one eigenvector of \(\mathbf { A }\) is \(\left( \begin{array} { r } 1 \\ - 2 \\ - 1 \end{array} \right)\), find the corresponding eigenvalue. Given also that another eigenvalue of \(\mathbf { A }\) is 4, find a corresponding eigenvector. Given further that \(\left( \begin{array} { r } 1 \\ - 4 \\ - 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\), with corresponding eigenvalue 1 , find matrices \(\mathbf { P }\) and \(\mathbf { Q }\), together with a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { A } ^ { 5 } = \mathbf { P D Q }\).

1 Given that 5 is an eigenvalue of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 3 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)$$ find a corresponding eigenvector. Hence find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf { A } + \mathbf { A } ^ { 2 }\).