4.03m det(AB) = det(A)*det(B)

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 \(P\)

3 The matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 2 & - 3 & 6 \\ 2 & 2 & - 4 \end{array} \right)\).

1. Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } + 6 \lambda ^ { 2 } - 9 \lambda - 14 = 0\).
2. Show that - 1 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
3. Find an eigenvector corresponding to the eigenvalue - 1 .
4. Verify that \(\left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\).
5. Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { M } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
6. Use the Cayley-Hamilton theorem to express \(\mathbf { M } ^ { - 1 }\) in the form \(a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)

3 Let \(\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k \\ 1 & 1 & 3 \\ 1 & 0 & - 1 \end{array} \right) (\) where \(k \neq 4 )\) and \(\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6 \\ - 1 & 3 & 1 \\ 1 & - 2 & - 2 \end{array} \right)\).

1. Find \(\mathbf { P } ^ { - 1 }\) in terms of \(k\), and show that, when \(k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4 \\ 4 & - 6 & - 10 \\ - 1 & 2 & 2 \end{array} \right)\).
2. Verify that \(\left( \begin{array} { l } 4 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\), and find the corresponding eigenvalues.
3. Show that \(\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10 \\ 2 & - 3 & - 5 \\ 0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4 \\ - 3 & 6 & 6 \\ 1 & - 2 & - 2 \end{array} \right)\). Section B (18 marks)

3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 7 & 3 \\ - 4 & - 1 \end{array} \right)\).

1. Find the eigenvalues, and corresponding eigenvectors, of the matrix \(\mathbf { M }\).
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\).
3. Given that \(\mathbf { M } ^ { n } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), show that \(a = - \frac { 1 } { 2 } + \frac { 3 } { 2 } \times 5 ^ { n }\), and find similar expressions for \(b , c\) and \(d\). Section B (18 marks)

9

1. Describe fully the transformation Q , represented by the matrix \(\mathbf { Q }\), where \(\mathbf { Q } = \left( \begin{array} { r l } 0 & 1 \\ - 1 & 0 \end{array} \right)\). The transformation M is represented by the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { r r } 0 & - 1 \\ 0 & 1 \end{array} \right)\).
2. M maps all points on the line \(y = 2\) onto a single point, P. Find the coordinates of P.
3. M maps all points on the plane onto a single line, \(l\). Find the equation of \(l\).
4. M maps all points on the line \(n\) onto the point ( - 6 , 6). Find the equation of \(n\).
5. Show that \(\mathbf { M }\) is singular. Relate this to the transformation it represents.
6. R is the composite transformation M followed by Q . R maps all points on the plane onto the line \(q\). Find the equation of \(q\).

8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1 \\ - 1 & 0 & - 3 \\ 1 & - 3 & 0 \end{array} \right)\). Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.

1. Show that \(| \mathbf { A } | = | \mathbf { B } |\).
2. Verify that \(| \mathbf { A B } | = | \mathbf { A } \| \mathbf { B } |\).
3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\).

5

1. Find the volume scale factor of the transformation with associated matrix \(\left( \begin{array} { r r r } 1 & 2 & 0 \\ 0 & 3 & - 1 \\ - 1 & 0 & 2 \end{array} \right)\).
2. The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf { P }\) and \(\mathbf { Q }\) respectively.
   1. Write down an expression for the associated matrix of the combined transformation S followed by T. The determinant of \(\mathbf { P }\) is 3 and \(\mathbf { Q } = \left( \begin{array} { r r } k & 3 \\ - 1 & 2 \end{array} \right)\), where \(k\) is a constant.
   2. Given that this combined transformation preserves both orientation and area, determine the value of \(k\).

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } \lambda & 2 \\ 2 & \lambda \end{array} \right)\) and \(\left( \begin{array} { c c } \mu & 1 \\ 1 & \mu \end{array} \right)\) respectively, where \(\lambda\) and \(\mu\) are constants.

1. Investigate whether \(\mathbf { M }\) and \(\mathbf { N }\) are commutative under multiplication.
2. You are now given that \(\mathbf { M N } = \mathbf { I }\).
   1. Write down a relationship between \(\operatorname { det } \mathbf { M }\) and \(\operatorname { det } \mathbf { N }\).
   2. Given that \(\lambda > 0\), find the exact values of \(\lambda\) and \(\mu\).
   3. Hence verify your answer to part (i).

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.

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.

1. Show that \(| \mathrm { A } | = | \mathrm { B } |\).
2. Verify that \(| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |\).
3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\). The \(2 \times 2\) matrix \(A\) represents a transformation \(T\) which has the following properties.
   The transformation \(S\) is represented by the matrix \(B\) where \(B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
   (b) Find the equation of the line of invariant points of S .
   (c) Show that any line of the form \(y = x + c\) is an invariant line of S .

$$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}$$

1. Find \(\mathbf{A}^{-1}\) [2 marks]
2. The determinant of \(\mathbf{A}^2\) is equal to 4. Find the possible values of \(k\). [3 marks]