4.03h Determinant 2x2: calculation

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 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 \(\mathbf { A B }\).

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 \(\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)\), 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 \(\mathbf { M }\).

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

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 \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 4 & 2 \\ 0 & - 1 & 1 \\ 0 & 0 & 2 \end{array} \right) .$$

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 eigenvalues \(b , - 1,1\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right)$$ respectively.
3. Find \(\mathbf { A }\) in terms of b.

9 It is given that \(a\) is a positive constant.

1. Show that the system of equations $$\begin{aligned} a x + ( 2 a + 5 ) y + ( a + 1 ) z & = 1 \\ - 4 y & = 2 \\ 3 y - z & = 3 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & 2 a + 5 & a + 1 \\ 0 & - 4 & 0 \\ 0 & 3 & - 1 \end{array} \right)$$
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , - 1\) and - 4 .
3. Find a matrix \(\mathbf { P }\) such that $$\mathbf { A } = \mathbf { P } \left( \begin{array} { r r r } a & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 4 \end{array} \right) \mathbf { P } ^ { - 1 } .$$
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right) .$$ Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }$$ where \(p\) and \(q\) are integers to be determined.

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 \(k\).

3. The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c } k + 5 & - 2 \\ - 3 & k \end{array} \right)$$

1. Determine the values of \(k\) for which \(\mathbf { M }\) is singular. Given that \(\mathbf { M }\) is non-singular,
2. find \(\mathbf { M } ^ { - 1 }\) in terms 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. Given that $$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).

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. Given that

$$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
   

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 \(O R S\).

5. \(\mathbf { A } = \left( \begin{array} { c c } a & - 5 \\ 2 & a + 4 \end{array} \right)\), where \(a\) is real.

1. Find \(\operatorname { det } \mathbf { A }\) in terms of \(a\).
2. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(a\). Given that \(a = 0\),
3. find \(\mathbf { A } ^ { - 1 }\).

8. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ - 1 & 3 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { A }\).
2. Find \(\mathbf { A } ^ { - 1 }\). The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\). Given that the area of triangle \(S\) is 72 square units,
3. find the area of triangle \(R\). The triangle \(S\) has vertices at the points \(( 0,4 ) , ( 8,16 )\) and \(( 12,4 )\).
4. Find the coordinates of the vertices of \(R\).

4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).

1. Find the coordinates of the vertices of \(T ^ { \prime }\).
2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
3. Find \(\mathbf { Q R }\).
4. Find the determinant of \(\mathbf { Q R }\).
5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).