4.03a Matrix language: terminology and notation

10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where \(p\) is a positive constant and \(p \neq 3\).

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.

16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where \(p\) is a positive constant and \(p \neq 3\).

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis. 11 Answer only one of the following two alternatives. \section*{EITHER} State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\).

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ 1 & 3 \end{array} \right)\).

1. Calculate \(\operatorname { det } \mathbf { A }\).
2. Write down \(\mathbf { A } ^ { - 1 }\).
3. Hence solve the equation \(\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }\).
4. Write down the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = 4 \mathbf { I }\). Matrices \(\mathbf { C }\) and \(\mathbf { D }\) are given by \(\mathbf { C } = \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)\) where \(p\) is a constant.
5. Find, in terms of \(p\),
   It is observed that \(\mathbf { C D } \neq \mathbf { D C }\).
6. The result that \(\mathbf { C D } \neq \mathbf { D C }\) is a counter example to the claim that matrix multiplication has a particular property. Name this property.

4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\)

1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
3. 1. Find the exact area of the region \(R ^ { \prime }\).
   2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.

3 You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 4 }\).
2. Describe the transformation that \(\mathbf { A }\) represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
3. Write down the matrix \(\mathbf { B }\). The point \(P\) has coordinates (2, 3, 4). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
4. Find the coordinates of \(P ^ { \prime }\).

1

1. The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l l l } 1 & 0 & - 2 & 2 \\ 4 & 2 & - 2 & 3 \end{array} \right)\).
   1. Write down the dimensions of \(\mathbf { P }\).
   2. Write down the transpose of \(\mathbf { P }\).
2. The matrices \(\mathbf { Q } , \mathbf { R }\) and \(\mathbf { S }\) are given by \(\mathbf { Q } = \left( \begin{array} { l l } 1 & 2 \end{array} \right) , \mathbf { R } = \left( \begin{array} { r r } 3 & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { S } = \left( \begin{array} { l l } 3 & - 2 \end{array} \right)\). Write down the sum of the two of these matrices which are conformable for addition.
3. The dimensions of matrix \(\mathbf { A }\) are 4 by 5. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are conformable for multiplication so that the matrix \(\mathbf { C } = \mathbf { B A }\) can be formed. The matrix \(\mathbf { C }\) has 6 rows.
   1. Write down the number of columns that \(\mathbf { C }\) has.
   2. Write down the dimensions of \(\mathbf { B }\).
   3. Explain whether the matrix \(\mathbf { A B }\) can be formed.
4. Find the value of \(c\) for which \(\left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right)\).

8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y \\ y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.

1. (a) The matrices \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d \\ d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right) \left( \begin{array} { c c } c & - d \\ d & c \end{array} \right) = \left( \begin{array} { c c } p & - q \\ q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
   (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.

5

1. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]\).
   Given that the image of the point \(( 5,2 )\) under the transformation represented by \(\mathbf { A }\) is \(( - 2,1 )\), find the value of \(c\) and the value of \(d\).
   [0pt] [4 marks]
2. The matrix \(\mathbf { B }\) is defined by \(\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]\).
   1. Show that \(\mathbf { B } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Describe the transformation represented by the matrix \(\mathbf { B }\) as a combination of two geometrical transformations.
   3. Find the matrix \(\mathbf { B } ^ { 17 }\). \(6 \quad \mathrm {~A}\) curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

1 The transformation R of the plane is reflection in the line \(x = 0\).

1. Write down the matrix \(\mathbf { M }\) associated with R .
2. Find \(\mathbf { M } ^ { 2 }\).
3. Interpret the result of part (b) in terms of the transformation \(R\).

1. $$\left( \begin{array} { l l } x & 9 \\ y & z \end{array} \right) - 3 \left( \begin{array} { l l } z & y \\ z & y \end{array} \right) = k \mathbf { I }$$ where \(x , y , z\) and \(k\) are constants.
Determine the value of \(x\), the value of \(y\) and the value of \(z\).

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.

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 \(l _ { 2 }\) .
   6. \(\mathbf { M } = \left( \begin{array} { c c c } 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), 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. Taking \(k = 3\) ,find cartesian equations of \(l _ { 2 }\) .

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)

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a triangle with vertices \(A ( 1,1 ) , B ( 3,1 )\) and \(C ( 3,2 )\). \includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}

1. The triangle \(D E F\) is obtained by applying to triangle \(A B C\) the transformation T represented by the matrix $$\left[ \begin{array} { r r } 2 & 2 \\ - 2 & 2 \end{array} \right]$$
   1. Calculate the coordinates of \(D , E\) and \(F\).
   2. Draw the triangle \(D E F\) on Figure 1.
2. Given that T is a combination of an enlargement and a rotation, find the exact value of:
   1. the scale factor of the enlargement;
   2. the magnitude of the angle of the rotation.

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.}

6 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left[ \begin{array} { c c } 5 & 2 \\ - 3 & 4 \end{array} \right]$$ 6

1. \(\quad\) Find \(\operatorname { det } \mathbf { A }\) 6
2. Find \(\mathbf { A } ^ { - 1 }\) 6
3. Given that \(\mathbf { A B } = \left[ \begin{array} { c c } 9 & 6 \\ 5 & 12 \end{array} \right]\) and \(\mathbf { M } = 2 \mathbf { A } + \mathbf { B }\) find the matrix \(\mathbf { M }\)

11 Prove by induction that, for all integers \(n \geq 1\), $$\left( \mathbf { A B A } ^ { - 1 } \right) ^ { n } = \mathbf { A B } ^ { n } \mathbf { A } ^ { - 1 }$$ where \(\mathbf { A }\) and \(\mathbf { B }\) are square matrices of equal dimensions, and \(\mathbf { A }\) is non-singular.

11 The Cartesian equation of the line \(L _ { 1 }\) is $$\frac { x + 1 } { 3 } = \frac { - y + 5 } { 2 } = \frac { 2 z + 5 } { 3 }$$ The Cartesian equation of the line \(L _ { 2 }\) is $$\frac { 2 x - 1 } { 2 } = \frac { y - 14 } { m } = \frac { z + 12 } { p }$$ The non-singular matrix \(\mathbf { N } = \left[ \begin{array} { c c c } - 0.5 & 1 & 2 \\ 1 & b & 4 \\ - 3 & - 2 & c \end{array} \right]\) maps the line \(L _ { 1 }\) onto the line \(L _ { 2 }\) Calculate the values of the constants \(b , c , m\) and \(p\) Fully justify your answers.

11

1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 5 } { 2 } & - \frac { 3 } { 2 } \\ - \frac { 3 } { 2 } & \frac { 13 } { 2 } \end{array} \right]$$ 11
2. (i) Describe how the directions of the invariant lines of the transformation represented by \(\mathbf { M }\) are related to each other. Fully justify your answer.
   [0pt] [2 marks]
   11 (b) (ii) Describe fully the transformation represented by \(\mathbf { M }\)

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 \(k\).

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 \(k\).