Describe single transformation from matrix

6.

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

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

1. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A }\).
2. Hence find the smallest positive integer value of \(n\) for which $$\mathbf { A } ^ { n } = \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. The transformation represented by the matrix \(\mathbf { A }\) followed by the transformation represented by the matrix \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\). Given that \(\mathbf { C } = \left( \begin{array} { r r } 2 & 4 \\ - 3 & - 5 \end{array} \right)\),
3. find the matrix \(\mathbf { B }\).

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

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\).
   
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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 \(C\).

2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1 \\ 5 & 2 \end{array} \right)$$

1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
3. write down \(\mathbf { C } ^ { 100 }\). \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}

3. (a) Given that $$\mathbf { A } = \left( \begin{array} { c c } 1 & \sqrt { } 2 \\ \sqrt { } 2 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { 2 }\),
2. describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
   (b) Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ describe fully the geometrical transformation represented by \(\mathbf { B }\).
   (c) Given that $$\mathbf { C } = \left( \begin{array} { c c } k + 1 & 12 \\ k & 9 \end{array} \right)$$ where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.

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

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents an enlargement, scale factor - 2 , with centre the origin.
2. Write down the matrix \(\mathbf { B }\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } 3 & k \\ - 2 & 3 \end{array} \right) , \quad \text { where } k \text { is a positive constant. }$$ Triangle \(T\) has an area of 16 square units. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of the triangle \(T ^ { \prime }\) is 224 square units, find the value of \(k\).

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

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(U\) maps the point \(A\), with coordinates \(( p , q )\), onto the point \(B\), with coordinates \(( 6 \sqrt { } 2,3 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\).
3. 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 \(\mathbf { R }\).
4. Find the matrix \(\mathbf { R }\).
5. Deduce that the transformation \(T\) is self-inverse.

1. $$\mathbf { R } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { S } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$

1. Find \(\mathbf { R } ^ { 2 }\).
2. Find \(\mathbf { R S }\).
3. Describe the geometrical transformation represented by \(\mathbf { R S }\).

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

1. Find \(\mathbf { R } ^ { 2 }\).
2. Describe the geometrical transformation represented by \(\mathbf { R } ^ { 2 }\).
3. Describe the geometrical transformation represented by \(\mathbf { R }\).

9

1. Write down the matrix, \(\mathbf { A }\), that represents an enlargement, centre ( 0,0 ), with scale factor \(\sqrt { 2 }\).
2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
3. Given that \(\mathbf { C } = \mathbf { A B }\), show that \(\mathbf { C } = \left( \begin{array} { r r } 1 & 1 \\ - 1 & 1 \end{array} \right)\).
4. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
5. Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).

7 Describe fully the geometrical transformation represented by each of the following matrices:

1. \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\),
2. \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\),
3. \(\left( \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right)\),
4. \(\left( \begin{array} { r r } 0.8 & 0.6 \\ - 0.6 & 0.8 \end{array} \right)\).

3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h] \end{figure}

1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).

6

1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).

5

1. Find the matrix that represents a reflection in the line \(y = - x\).
2. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 4 \end{array} \right)\).
   1. Describe fully the geometrical transformation represented by \(\mathbf { C }\).
   2. State the value of the determinant of \(\mathbf { C }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { C }\).

5

1. Write down the matrix that represents a reflection in the line \(y = x\).
2. Describe fully the geometrical transformation represented by each of the following matrices: $$\begin{aligned} & \text { (i) } \left( \begin{array} { c c } 5 & 0 \\ 0 & 1 \end{array} \right) \text {, } \\ & \text { (ii) } \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } \sqrt { 3 } \\ - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \text {. } \end{aligned}$$

4

1. Find the matrix that represents a shear with the \(y\)-axis invariant, the image of the point \(( 1,0 )\) being the point \(( 1,4 )\).
2. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\).
   1. Describe fully the geometrical transformation represented by \(\mathbf { X }\).
   2. Find the value of the determinant of \(\mathbf { X }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { X }\).

1 Transformation A is represented by matrix \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and transformation B is represented by matrix \(\mathbf { B } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)\).

1. Describe transformations A and B .
2. Find the matrix for the composite transformation A followed by B .

9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.

7

1. The transformation T is defined by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]$$
   1. Describe the transformation T geometrically.
   2. Calculate the matrix product \(\mathbf { A } ^ { 2 }\).
   3. Explain briefly why the transformation T followed by T is the identity transformation.
2. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$$
   1. Calculate \(\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }\).
   2. Calculate \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).

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

1. Calculate:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { B A }\).
2. Describe fully the geometrical transformation represented by each of the following matrices:
   1. \(\mathbf { A }\);
   2. \(\mathbf { B }\);
   3. \(\mathbf { B A }\).

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$ Describe fully the geometrical transformation represented by each of the following matrices:

1. A ;
2. B ;
3. \(\quad \mathbf { A } ^ { 2 }\);
4. \(\quad \mathbf { B } ^ { 2 }\);
5. AB.
   □

7 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$

1. 1. Calculate the matrix \(\mathbf { A } ^ { 2 }\).
   2. Show that \(\mathbf { A } ^ { 3 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
   1. \(\mathrm { A } ^ { 3 }\);
   2. A.