4.03i Determinant: area scale factor and orientation

6 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right)\).

1. The transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { - 1 }\) transforms a triangle of area \(30 \mathrm {~cm} ^ { 2 }\) into a triangle of area \(d \mathrm {~cm} ^ { 2 }\). Find the value of \(d\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 2 ^ { n } - 1 & 1 \end{array} \right)$$
3. The line \(y = 2 x\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\), where \(\mathbf { B } = \left( \begin{array} { r l } 1 & 0 \\ 33 & 0 \end{array} \right)\). Find the value of \(n\).

4 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a rotation of \(60 ^ { \circ }\) anticlockwise about the origin followed by a one-way stretch in the \(x\)-direction, scale factor \(d ( d \neq 0 )\).

1. Find \(\mathbf { M }\) in terms of \(d\).
2. The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto a parallelogram of area \(\frac { 1 } { 2 } d ^ { 2 }\) units \({ } ^ { 2 }\). Show that \(d = 2\).
   The matrix \(\mathbf { N }\) is such that \(\mathbf { M N } = \left( \begin{array} { l l } 1 & 1 \\ \frac { 1 } { 2 } & \frac { 1 } { 2 } \end{array} \right)\).
3. Find \(\mathbf { N }\).
4. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M N }\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 } \\ \mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)\), where \(a , b , c\) are real constants and \(b \neq 0\).

1. Show that \(\mathbf { M }\) does not represent a rotation about the origin.
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
   It is given that \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( 5,1 )\).
3. Find \(\mathbf { M }\).
4. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(12 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \left( \begin{array} { c c } 14 & 0 \\ 0 & 1 \end{array} \right)\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details 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. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Given that the area of triangle \(D E F\) is \(28 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(A B C\).

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

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied. [4]
2. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\) .
3. Given that the area of triangle \(P Q R\) is \(35 \mathrm {~cm} ^ { 2 }\) ,find the area of triangle \(D E F\) .

5 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r }

8

1. Find the values of \(a\) for which the system of equations $$\begin{aligned} 3 x + y + z & = 0 \\ a x + 6 y - z & = 0 \\ a y - 2 z & = 0 \end{aligned}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 1 \\ 0 & 6 & - 1 \\ 0 & 0 & - 2 \end{array} \right) .$$
2. Use the characteristic equation of \(\mathbf { A }\) to find the inverse of \(\mathbf { A } ^ { 2 }\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) do not intersect and are both perpendicular to \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The line \(l\) intersects \(\Pi _ { 1 }\) at the point \(( 1,6,0 )\) and intersects \(\Pi _ { 2 }\) at the point \(( 3 , - 6,0 )\).

1. Find Cartesian equations of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
2. Express the vector equation of \(l\) in the form \(\left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be determined, and hence show that for points on \(l , \frac { 1 } { 2 } x + \frac { 1 } { 12 } y = 1\) and \(z = 0\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-16_2715_40_144_2008}
3. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 3 \lambda ^ { 2 } + \frac { 7 } { 4 } \lambda = 0\) and hence find the eigenvalues of \(\mathbf { A }\). The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 3 \\ 1 & 2 & 3 \\ \frac { 1 } { 2 } & \frac { 1 } { 12 } & 0 \end{array} \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-17_194_1711_484_212}
4. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\), where \(n\) is a positive integer. [6] \includegraphics[max width=\textwidth, alt={}]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_65_1581_335_322} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_72_1579_511_324} \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_2718_35_144_2012} If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } - 6 & 2 & 13 \\ 0 & - 2 & 5 \\ 0 & 0 & 8 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

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

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 non-zero eigenvalues \(a , \frac { 1 } { 2 } , 2\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right) ,$$ respectively.
3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 11 & 1 & 8 \\ 0 & - 2 & 0 \\ - 16 & 1 & 13 \end{array} \right)$$

1. Show that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue.
2. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 19 \lambda - 30 = 0\) and hence find the other eigenvalues of \(\mathbf { A }\). \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-08_2717_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-09_2726_33_97_22}
3. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

3. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 4 \\ 1 & 1 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular. The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\).
   Given that the area of triangle \(R\) is 10 square units,
2. find the area of triangle \(S\). Given that $$\mathbf { B } = \mathbf { A } ^ { 4 }$$ and that the triangle \(R\) is transformed to the triangle \(T\) by the matrix \(\mathbf { B }\),
3. find, without evaluating \(\mathbf { B }\), the area of triangle \(T\).

6.

1. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
   1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
   2. Describe fully the single transformation represented by the matrix \(\mathbf { B }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
   3. Find \(\mathbf { C }\).
   4. \(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 \\ 1 & k \end{array} \right)\), where \(k\) is a real number. Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).

7. The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 5 \\ - 3 & 2 \end{array} \right)$$ The transformation represented by \(\mathbf { A }\) maps triangle \(T\) onto triangle \(T ^ { \prime }\) Given that the area of triangle \(T\) is \(23 \mathrm {~cm} ^ { 2 }\)

1. determine the area of triangle \(T ^ { \prime }\) (2) The point \(P\) has coordinates ( \(3 p + 2,2 p - 1\) ) where \(p\) is a constant. The transformation represented by \(\mathbf { A }\) maps \(P\) onto the point \(P ^ { \prime }\) with coordinates \(( 17 , - 18 )\)
2. Determine the value of \(p\). Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
3. describe fully the single geometrical transformation represented by matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { C }\) is equivalent to the transformation represented by matrix \(\mathbf { B }\)
4. Determine C \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-21_2255_50_314_34}
   

4. $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. Describe the single geometrical transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(210 ^ { \circ }\) anticlockwise about centre \(( 0,0 )\).
2. Write down the matrix \(\mathbf { B }\), giving each element in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
3. Find \(\mathbf { C }\). The hexagon \(H\) is transformed onto the hexagon \(H ^ { \prime }\) by the matrix \(\mathbf { C }\).
4. Given that the area of hexagon \(H\) is 5 square units, determine the area of hexagon \(H ^ { \prime }\)

8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a \\ 4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$

1. Find the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(a\).
   (3) The matrix \(\mathbf { P }\) represents the transformation \(U\) which transforms a triangle \(T _ { 1 }\) onto the triangle \(T _ { 2 }\).
   The triangle \(T _ { 2 }\) has vertices at the points ( \(- 3 a , - 4 a\) ), ( \(6 a , 8 a\) ), and ( \(- 20 a , 15 a\) ).
2. Find the coordinates of the vertices of \(T _ { 1 }\)
3. Hence, or otherwise, find the area of triangle \(T _ { 2 }\) in terms of \(a\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through an angle \(\alpha\) clockwise about the origin, where \(\tan \alpha = \frac { 4 } { 3 }\) and \(0 < \alpha < \frac { \pi } { 2 }\)
4. Write down the matrix \(\mathbf { Q }\), giving each element as an exact value. The transformation \(U\) followed by the transformation \(V\) is the transformation \(W\). The matrix \(\mathbf { R }\) represents the transformation \(W\).
5. Find the matrix \(\mathbf { R }\).

10. In your answers to this question, the elements of each matrix should be expressed in exact form in surds where necessary. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(45 ^ { \circ }\) anticlockwise about the origin.

1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(60 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix \(\mathbf { Q }\). The transformation \(U\) followed by the transformation \(V\) is the transformation \(T\). The transformation \(T\) is represented by the matrix \(\mathbf { R }\).
3. Use your matrices from parts (a) and (b) to find the matrix \(\mathbf { R }\).
4. Give a full geometric description of \(T\) as a single transformation.
5. Deduce from your answers to parts (c) and (d) that \(\sin 75 ^ { \circ } = \frac { 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }\) and find the
   exact value of \(\cos 75 ^ { \circ }\), explaining your answers fully.

6. (i) $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(45 ^ { \circ }\) clockwise about the origin.
2. Write down the matrix \(\mathbf { B }\), giving each element of the matrix in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
3. Determine \(\mathbf { C }\).
   (ii) The trapezium \(T\) has vertices at the points \(( - 2,0 ) , ( - 2 , k ) , ( 5,8 )\) and \(( 5,0 )\), where \(k\) is a positive constant. Trapezium \(T\) is transformed onto the trapezium \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 5 & 1 \\ - 2 & 3 \end{array} \right)$$ Given that the area of trapezium \(T ^ { \prime }\) is 510 square units, calculate the exact value of \(k\).

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

7. $$A = \left( \begin{array} { c c } - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Determine the matrix \(\mathbf { A } ^ { 2 }\)
2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\)
3. Hence determine the smallest positive integer value of \(n\) for which \(\mathbf { A } ^ { n } = \mathbf { I }\) The matrix \(\mathbf { B }\) represents a stretch scale factor 4 parallel to the \(x\)-axis.
4. Write down the matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\)
5. Determine the matrix \(\mathbf { C }\) The parallelogram \(P\) is transformed onto the parallelogram \(P ^ { \prime }\) by the matrix \(\mathbf { C }\)
6. Given that the area of parallelogram \(P ^ { \prime }\) is 20 square units, determine the area of parallelogram \(P\)

1. (i) \(\mathbf { A } = \left( \begin{array} { c c } - 3 & 8 \\ - 3 & k \end{array} \right) \quad\) where \(k\) is a constant The transformation represented by \(\mathbf { A }\) transforms triangle \(T\) to triangle \(T ^ { \prime }\) The area of triangle \(T ^ { \prime }\) is three times the area of triangle \(T\)

Determine the possible values of \(k\) (ii) \(\mathbf { B } = \left( \begin{array} { r r } a & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B C } = \left( \begin{array} { l l l } 2 & 5 & 1 \\ 1 & 4 & 2 \end{array} \right)\) where \(a\) is a constant Determine, in terms of \(a\), the matrix \(\mathbf { C }\)

1. (a) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } 1 & r \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & \left( 2 ^ { n } - 1 \right) r \\ 0 & 2 ^ { n } \end{array} \right)$$ where \(r\) is a constant. $$\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 5 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 2 \end{array} \right) ^ { 4 }$$ The transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\) is represented by the matrix \(\mathbf { B }\) (b) (i) Determine \(\mathbf { N }\) in the form \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) where \(a , b , c\) and \(d\) are integers.
(ii) Determine B Hexagon \(S\) is transformed onto hexagon \(S ^ { \prime }\) by matrix \(\mathbf { B }\) (c) Given that the area of \(S ^ { \prime }\) is 720 square units, determine the area of \(S\)

10. $$\mathbf { A } = \left( \begin{array} { c c } 3 \sqrt { } 2 & 0 \\ 0 & 3 \sqrt { } 2 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 0 & 1 \\ 1 & 0 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & - \frac { 1 } { \sqrt { } 2 } \\ \frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$

1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
2. Find \(\mathbf { D }\).
3. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3 \\ 3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
4. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
5. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).

9. $$\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$

1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\).
3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
5. Find the coordinates of \(C\).