4.03i Determinant: area scale factor and orientation

P, Q and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P.

1. Write down the matrix A. [1]

Q is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). B is the matrix that represents Q.

1. Find the matrix B. [2]

T is P followed by Q. C is the matrix that represents T.

1. Determine the matrix C. [2]

\(L\) is the line whose equation is \(y = x\).

1. Explain whether or not \(L\) is a line of invariant points under T. [2]

An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).

1. Explain what the value of the determinant of C means about
   [3]

A transformation T is represented by the matrix \(\mathbf{T}\) where \(\mathbf{T} = \begin{pmatrix} x^2 + 1 & -4 \\ 3 - 2x^2 & x^2 + 5 \end{pmatrix}\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q'\). Find the smallest possible value of the area of \(Q'\). [5]

The \(2 \times 2\) matrix \(\mathbf{A}\) 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 \(\mathbf{A}\). [8]

The transformation \(S\) is represented by the matrix \(\mathbf{B}\) where \(\mathbf{B} = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}\).

1. Find the equation of the line of invariant points of \(S\). [2]
2. Show that any line of the form \(y = x + c\) is an invariant line of \(S\). [3]

The triangle \(T\) has vertices at the points \((1, k)\), \((3,0)\) and \((11,0)\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T'\) by the matrix $$\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix}$$ Given that the area of triangle \(T'\) is 364 square units, find the value of \(k\). [4]

The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$

1. Determine
   1. the value of \(k\),
   2. the smallest value of \(\theta\) [4]

A square \(S\) has vertices at the points with coordinates \((0, 0)\), \((a, -a)\), \((2a, 0)\) and \((a, a)\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S'\) by the transformation represented by \(\mathbf{M}\).

1. Determine, in terms of \(a\), the area of \(S'\) [2]

You are given that \(M = \begin{pmatrix} \frac{1}{\sqrt{3}} & -\sqrt{3} \\ \frac{1}{\sqrt{3}} & 1 \end{pmatrix}\).

1. Show that \(M\) is non-singular. [2]

The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(M\). Given that the area of hexagon \(R\) is 5 square units,

1. find the area of hexagon \(S\). [1]

The matrix \(M\) 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 \((0,0)\).

1. Find the value of \(k\). [2]
2. Find the value of \(\theta\). [2]

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

1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin. [5]
2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines. [3]

$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

1. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]

The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).

1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]

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]

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 }