4.03i Determinant: area scale factor and orientation

8 It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector
e. Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector. Deduce that \(\lambda + \lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } + \mathbf { A } ^ { - 1 }\). It is given that 1 is an eigenvalue of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ Find a corresponding eigenvector. It is also given that \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) are eigenvectors of the matrix \(\mathbf { A }\). Find the corresponding eigenvalues.
Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$

10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) has an eigenvector \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\). Find the corresponding eigenvalue. The matrix \(\mathbf { A }\) also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix \(\mathbf { P }\) such that \(\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\), where \(\mathbf { D }\) is a diagonal matrix which is to be determined. The matrix \(\mathbf { B }\) is such that \(\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }\), where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5 \\ 1 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right)$$ By using the expression \(\mathbf { P D P } ^ { - 1 }\) for \(\mathbf { A }\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf { B }\).
[0pt] [Question 11 is printed on the next page.]

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).

The vector \(\mathbf { e }\) is an eigenvector of each of the \(n \times n\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Prove that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with eigenvalue \(\lambda \mu\). It is given that the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 2 & 2 \\ - 2 & - 2 & - 2 \\ 1 & 2 & 2 \end{array} \right) ,$$ has eigenvectors \(\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 0 \\ - 1 \end{array} \right)\). Find the corresponding eigenvalues. Given that 2 is also an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 2 & 2 \\ 2 & 2 & 2 \\ - 3 & - 6 & - 6 \end{array} \right) ,$$ has the same eigenvectors as \(\mathbf { A }\). Given that \(\mathbf { A B } = \mathbf { C }\), find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { C } ^ { 2 } \mathbf { P } = \mathbf { D }$$

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 \(2 \times 2\) 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 .

5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\).

1. The diagram in the Printed Answer Booklet shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. Find the equation of the line of invariant points of this transformation.
3. (a) Find the determinant of \(\mathbf { M }\).
   (b) Describe briefly how this value relates to the transformation represented by \(\mathbf { M }\).

10 You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
3. For each of the values of \(a\) found in part (b) determine whether the equations have

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.

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4 \\ - 0.8 & 1.8 \end{array} \right)\).

1. Find \(\operatorname { det } \mathbf { A }\). The matrix A represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
2. By considering the determinants of these transformations, determine the scale factor of the stretch.
3. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
4. Find the angle of rotation.

6 In this question you must show detailed reasoning.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).

1. Define the transformation represented by \(\mathbf { A }\).
2. Show that the area of any object shape is invariant under the transformation represented by \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r l } 7 & 2 \\ 21 & 7 \end{array} \right)\). You are given that \(\mathbf { B }\) represents the transformation which is the result of applying the following three transformations in the given order.

6

1. Using surd forms, find the matrix of a rotation about the origin through \(135 ^ { \circ }\) anticlockwise.
2. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { r r } - 1 & - 1 \\ 1 & - 1 \end{array} \right]\).
   1. Given that \(\mathbf { M }\) represents an enlargement followed by a rotation, find the scale factor of the enlargement and the angle of the rotation.
   2. The matrix \(\mathbf { M } ^ { 2 }\) also represents an enlargement followed by a rotation. State the scale factor of the enlargement and the angle of the rotation.
   3. Show that \(\mathbf { M } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   4. Deduce that \(\mathbf { M } ^ { 2012 } = - 2 ^ { n } \mathbf { I }\) for some positive integer \(n\).

6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k \\ 1 & \lambda - k \end{array} \right)\), and \(\lambda\) and \(k\) are real constants. and \(k\) are real constants.

1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\).
   2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
   Find the possible values of \(\lambda\).

8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta \\ \sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).

1. Show that T leaves areas unchanged for all values of \(\theta\).
2. Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
3. 1. Find \(\mathbf { M N } ^ { - 1 }\).
   2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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8

1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
   1. Find \(\mathbf { M } ^ { 2 }\).
   2. Write down the transformation represented by \(\mathbf { M }\).
   3. Hence state the geometrical significance of the result of part (i).
2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.

6 A transformation T of the plane has associated matrix \(\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1 \\ \lambda - 1 & - 1 \end{array} \right)\), where \(\lambda\) is a non-zero
constant.

1. 1. Show that T reverses orientation.
   2. State, in terms of \(\lambda\), the area scale factor of T .
2. 1. Show that \(\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }\).
   2. Hence specify the transformation equivalent to two applications of T .
3. In the case where \(\lambda = 1 , \mathrm {~T}\) is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
   1. Determine the matrix associated with S .
   2. Hence describe the transformation S .

4

1. A transformation with associated matrix \(\left( \begin{array} { r r r } m & 2 & 1 \\ 0 & 1 & - 2 \\ 2 & 0 & 3 \end{array} \right)\), where \(m\) is a constant, maps the vertices of a cube to points that all lie in a plane. Find \(m\).
2. The transformations S and T of the plane have associated matrices \(\mathbf { M }\) and \(\mathbf { N }\) respectively, where \(\mathbf { M } = \left( \begin{array} { r r } k & 1 \\ - 3 & 4 \end{array} \right)\) and the determinant of \(\mathbf { N }\) is \(3 k + 1\). The transformation \(U\) is equivalent to the combined transformation consisting of S followed by T . Given that U preserves orientation and has an area scale factor 2, find the possible values of \(k\).

8

1. Specify fully the transformation T of the plane associated with the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)\) and \(\lambda\) is a non-zero constant.
2. 1. Find detM.
   2. Deduce two properties of the transformation T from the value of detM.
3. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)\), where \(n\) is a positive integer.
4. Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T.

9 A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2 \\ \lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant. constant.

1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.

1. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5 \\ 2 & - 7 \end{array} \right)$$

1. Show that the matrix \(\mathbf { M }\) is non-singular. The transformation \(T\) of the plane is represented by the matrix \(\mathbf { M }\).
   The triangle \(R\) is transformed to the triangle \(S\) by the transformation \(T\).
   Given that the area of \(S\) is 63 square units,
2. find the area of \(R\).
3. Show that the line \(y = 2 x\) is invariant under the transformation \(T\).

5. $$\mathbf { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 } \\ \sqrt { 3 } & 1 \end{array} \right)$$

1. Show that \(\mathbf { M }\) is non-singular. The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of hexagon \(R\) is 5 square units,
2. find the area of hexagon \(S\). The matrix \(\mathbf { M }\) represents an enlargement, with centre \(( 0,0 )\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
3. Find the value of \(k\).
4. Find the value of \(\theta\).

1. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1 \\ k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,

1. determine
   1. the value of \(k\)
   2. the eigenvalue.
2. Hence determine a Cartesian equation of the invariant line under \(T\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$\mathbf { A } = \left( \begin{array} { r r } 3 & k \\ - 5 & 2 \end{array} \right)$$ where \(k\) is a constant.
Given that there exists a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix where $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { r r } 8 & 0 \\ 0 & - 3 \end{array} \right)$$

1. show that \(k = - 6\)
2. determine a suitable matrix \(\mathbf { P }\)