Transformation mapping problems

2. $$\mathbf { T } = \left( \begin{array} { l l l } 2 & 3 & 7 \\ 3 & 2 & 6 \\ a & 4 & b \end{array} \right) \quad \mathbf { U } = \left( \begin{array} { r r r } 6 & - 1 & - 4 \\ 15 & c & - 9 \\ - 8 & a & 5 \end{array} \right)$$ where \(a\), \(b\) and \(c\) are constants.
Given that \(\mathbf { T U } = \mathbf { I }\)

1. determine the value of \(a\), the value of \(b\) and the value of \(c\) The transformation represented by the matrix \(\mathbf { T }\) transforms the line \(l _ { 1 }\) to the line \(l _ { 2 }\) Given that \(l _ { 2 }\) has equation $$\frac { x - 1 } { 3 } = \frac { y } { - 4 } = z + 2$$
2. determine a Cartesian equation for \(l _ { 1 }\)

6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{array} \right) \quad a \neq 1$$

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
   . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
2. Find a vector equation for the line \(l _ { 1 }\)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 0 \\ 0 & 1 & 4 \\ 3 & - 2 & - 3 \end{array} \right)$$

1. Determine \(\mathbf { M } ^ { - 1 }\) The transformation represented by \(\mathbf { M }\) maps the plane \(\Pi _ { 1 }\) to the plane \(\Pi _ { 2 }\) The point \(( x , y , z )\) on \(\Pi _ { 1 }\) maps to the point \(( u , v , w )\) on \(\Pi _ { 2 }\)
2. Determine \(x , y\) and \(z\) in terms of \(u , v\) and \(w\) as appropriate. The plane \(\Pi _ { 1 }\) has equation $$3 x - 7 y + 2 z = - 3$$
3. Find a Cartesian equation for \(\Pi _ { 2 }\) Give your answer in the form \(a u + b v + c w = d\) where \(a , b , c\) and \(d\) are integers to be determined.

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For the eigenvalue 4, find a corresponding eigenvector. The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\). The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
3. Find a vector equation for the line \(l _ { 2 }\).

6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & 0 \\ 2 & - 2 & 1 \\ - 4 & 1 & - 1 \end{array} \right) , k \in \mathbb { R } , k \neq \frac { 1 } { 2 }$$

1. Show that \(\operatorname { det } \mathbf { M } = 1 - 2 k\).
2. Find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix $$\left( \begin{array} { r r r } 1 & 0 & 0 \\ 2 & - 2 & 1 \\ - 4 & 1 & - 1 \end{array} \right)$$ Given that \(l _ { 2 }\) has cartesian equation $$\frac { x - 1 } { 5 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 1 }$$
3. find a cartesian equation of the line \(l _ { 1 }\)

3 You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { 4 }\).
2. Describe the transformation that \(\mathbf { A }\) represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
3. Write down the matrix \(\mathbf { B }\). The point \(P\) has coordinates (2, 3, 4). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
4. Find the coordinates of \(P ^ { \prime }\).

1. (i)

$$\mathbf { P } = \left( \begin{array} { r r r } k & - 2 & 7 \\ - 3 & - 5 & 2 \\ k & k & 4 \end{array} \right)$$ where \(k\) is a constant Show that \(\mathbf { P }\) is non-singular for all real values of \(k\).
(ii) $$\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ - 3 & 0 \end{array} \right)$$ The matrix \(\mathbf { Q }\) represents a linear transformation \(T\) Under \(T\), the point \(A ( a , 2 )\) and the point \(B ( 4 , - a )\), where \(a\) is a constant, are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\) respectively. Given that the distance \(A ^ { \prime } B ^ { \prime }\) is \(\sqrt { 58 }\), determine the possible values of \(a\).

1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about } \\ \text { the } z \text {-axis is represented by the matrix } \\ \qquad \left( \begin{array} { c c c } \cos \theta & - \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array} \right) \end{array} \right]\)

Given that the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { c c c } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } & 0 \\ - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)$$ represents a rotation through \(\alpha ^ { \circ }\) anticlockwise about the \(z\)-axis with respect to the right-hand rule,

1. determine the value of \(\alpha\).
2. Hence determine the smallest possible positive integer value of \(k\) for which \(\mathbf { M } ^ { k } = \mathbf { I }\) The \(3 \times 3\) matrix \(\mathbf { N }\) represents a reflection in the plane with equation \(y = 0\)
3. Write down the matrix \(\mathbf { N }\). The point \(A\) has coordinates (-2, 4, 3)
   The point \(B\) is the image of the point \(A\) under the transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\).
4. Show that the coordinates of \(B\) are \(( 2 + \sqrt { 3 } , 2 \sqrt { 3 } - 1,3 )\) Given that \(O\) is the origin,
5. show that, to 3 significant figures, the size of angle \(A O B\) is \(66.9 ^ { \circ }\)
6. Hence determine the area of triangle \(A O B\), giving your answer to 3 significant figures.

The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} k & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}, \quad k \neq 1$$

1. Show that \(\det \mathbf{M} = 2 - 2k\). [2]
2. Find \(\mathbf{M}^{-1}\), in terms of \(k\). [5] The straight line \(l_1\) is mapped onto the straight line \(l_2\) by the transformation represented by the matrix \(\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}\). The equation of \(l_2\) is \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\), where \(\mathbf{a} = 4\mathbf{i} + \mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
3. Find a vector equation for the line \(l_1\). [5]

$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$

1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]

The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).

1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]

You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation \(R\) which is a reflection in 3-D.

1. State the plane of reflection of \(R\). [1]
2. Determine the value of \(a\). [3]
3. With reference to \(R\) explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3 \times 3\) identity matrix. [2]