4.03g Invariant points and lines

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 \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } k & 0 & 2 \\ 0 & - 1 & - 1 \\ 1 & 1 & - k \end{array} \right)$$ where \(k\) is a real constant.

1. Show that \(\mathbf { A }\) is non-singular.
   The matrices \(\mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 3 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 3 & - 1 & 1 \\ 1 & 1 & 2 \end{array} \right)$$ It is given that \(\mathbf { C A B } = \left( \begin{array} { l l } - 2 & - \frac { 3 } { 2 } \\ - 1 & - \frac { 3 } { 2 } \end{array} \right)\).
2. Find the value of \(k\).
3. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).

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 matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } 2 & k & k \\ 5 & - 1 & 3 \\ 1 & 0 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \quad \mathbf { C } = \left( \begin{array} { r c c } 0 & 1 & 1 \\ - 1 & 2 & 0 \end{array} \right)$$ where \(k\) is a real constant.

1. Find \(\mathbf { C A B }\).
2. Given that \(\mathbf { A }\) is singular, find the value of \(k\).
3. Using the value of \(k\) from part (b), find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).

5 Let \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 0 & 1 \end{array} \right)\), where \(a\) is a positive constant.

1. State the type of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \mathrm { na } \\ 0 & 1 \end{array} \right)$$ Let \(\mathbf { B } = \left( \begin{array} { c c } b & b \\ a ^ { - 1 } & a ^ { - 1 } \end{array} \right)\), where \(b\) is a positive constant.
3. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\).

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 \(\mathbf { M } = \left( \begin{array} { r r } \cos 2 \theta & - \sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{array} \right) \left( \begin{array} { l l } 1 & k \\ 0 & 1 \end{array} \right)\), where \(0 < \theta < \pi\) and \(k\) is a non-zero constant. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations, one of which is a shear.

1. Describe fully the other transformation and state the order in which the transformations are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find, in terms of \(k\), the value of \(\tan \theta\) for which \(\mathbf { M - I }\) is singular.
4. Given that \(k = 2 \sqrt { 3 }\) and \(\theta = \frac { 1 } { 3 } \pi\), show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(3 y + \sqrt { 3 } x = 0\).

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 \(D E F\) .

1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & b \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } a & 0 \\ 0 & 1 \end{array} \right)\), where \(a\) and \(b\) are positive constants.

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
   The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
2. Find, in terms of \(a\) and \(b\), the matrix which transforms parallelogram \(O P Q R\) onto the unit square.
   It is given that the area of \(O P Q R\) is \(2 \mathrm {~cm} ^ { 2 }\) and that the line \(\mathrm { x } + 3 \mathrm { y } = 0\) is invariant under the transformation represented by \(\mathbf { M }\).
3. Find the values of \(a\) and \(b\).

4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right)$$

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { B }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A B }\) onto triangle \(P Q R\).
3. Show that the triangles \(D E F\) and \(P Q R\) have the same area.
4. Find the matrix which transforms triangle \(P Q R\) onto triangle \(D E F\).
5. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A B }\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
2. Find the values of \(\theta\), for \(0 \leqslant \theta \leqslant \pi\), for which the transformation represented by \(\mathbf { M }\) has exactly one invariant line through the origin, giving your answers in terms of \(\pi\).

1

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by the matrix \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\). Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 4 \\ 2 & 2 \end{array} \right)\).
2. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(13 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
3. Find the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = \left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\).
4. Show that the origin is the only invariant point of the transformation in the \(x - y\) plane represented by \(\mathbf { A }\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 1 & 0 \\ 0 & k \end{array} \right) \left( \begin{array} { c c } 1 & 0 \\ k & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) or 1 .

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type 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. Show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(\mathrm { y } = \frac { \mathrm { k } ^ { 2 } } { 1 - \mathrm { k } } \mathrm { x }\). [4]
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find the value of \(k\) for which the area of triangle \(D E F\) is equal to the area of triangle \(A B C\).

5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.

1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).
3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
   Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
   The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
2. Find, in terms of \(k\), the area of parallelogram \(O P Q R\) and the matrix which transforms \(O P Q R\) onto the unit square.
3. Show that the line through the origin with gradient \(\frac { 1 } { k - 1 }\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).

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

1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-08_2715_31_106_2016} Let \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).
3. Give full details of the transformation represented by \(\mathbf { M }\).
4. Find the matrix \(\mathbf { N }\) such that \(\mathbf { N M } = \mathbf { C A B }\).

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

3

1. Given the matrix \(\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{array} \right)\) (where \(k \neq 3\) ), find \(\mathbf { Q } ^ { - 1 }\) in terms of \(k\).
   Show that, when \(k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1 \\ 1 & - 8 & 2 \\ 1 & - 5 & 1 \end{array} \right)\). The matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right)\), with corresponding eigenvalues \(1 , - 1\) and 3 respectively.
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\), and hence find the matrix \(\mathbf { M }\).
3. Write down the characteristic equation for \(\mathbf { M }\), and use the Cayley-Hamilton theorem to find integers \(a , b\) and \(c\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)

3

1. 1. A \(3 \times 3\) matrix \(\mathbf { M }\) has characteristic equation $$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$ Show that \(\lambda = 2\) is an eigenvalue of \(\mathbf { M }\). Find the other eigenvalues.
   2. An eigenvector corresponding to \(\lambda = 2\) is \(\left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\). Evaluate \(\mathbf { M } \left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\) and \(\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1 \\ - 1 \\ \frac { 1 } { 3 } \end{array} \right)\).
      Solve the equation \(\mathbf { M } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\).
   3. Find constants \(A , B , C\) such that $$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
2. A \(2 \times 2\) matrix \(\mathbf { N }\) has eigenvalues -1 and 2, with eigenvectors \(\binom { 1 } { 2 }\) and \(\binom { - 1 } { 1 }\) respectively. Find \(\mathbf { N }\). Section B (18 marks)

3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).

3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1 \\ 2 & 2 \end{array} \right)\).

9 The matrix \(\mathbf { M }\) represents the transformation T and is given by $$\mathbf { M } = \left[ \begin{array} { c c } 3 p + 1 & 12 \\ p + 2 & p ^ { 2 } - 3 \end{array} \right]$$ 9

1. In the case when \(p = 0\) show that the image of the point \(( 4,5 )\) under T is the point \(( 64 , - 7 )\) 9
2. In the case when \(p = - 2\) find the gradient of the line of invariant points under \(T\) 9
3. Show that \(p = 3\) is the only real value of \(p\) for which \(\mathbf { M }\) is singular.
   The curve \(C\) has equation $$y = \frac { 3 x ^ { 2 } + m x + p } { x ^ { 2 } + p x + m }$$ where \(m\) and \(p\) are integers.
   The vertical asymptotes of \(C\) are \(x = - 4\) and \(x = - 1\) The curve \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-12_867_1102_733_463}

8 The \(2 \times 2\) matrix A represents a transformation T which has the following properties.

• The image of the point \(( 0,1 )\) is the point \(( 3,4 )\).
• An object shape whose area is 7 is transformed to an image shape whose area is 35 .
• T has a line of invariant points.
  1. Find a possible matrix for \(\mathbf { A }\).

The transformation S is represented by the matrix \(\mathbf { B }\) where \(\mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).

Find the equation of the line of invariant points of S .

Show that any line of the form \(y = x + c\) is an invariant line of S .