4.03g Invariant points and lines

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$$

1. Prove by induction that, for all integers \(n \geq 1\), $$\mathbf{A}^n = \begin{bmatrix} 1 & 3^n - 1 \\ 0 & 3^n \end{bmatrix}$$ [4 marks]
2. Find all invariant lines under the transformation matrix \(\mathbf{A}\). Fully justify your answer. [6 marks]
3. Find a line of invariant points under the transformation matrix \(\mathbf{A}\). [2 marks]

The matrix \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) represents a transformation. Which one of the points below is an invariant point under this transformation? Circle your answer. [1 mark] \((1, 1) \quad (0, 2) \quad (3, 0) \quad (2, 1)\)

The matrix \(\mathbf{C}\) is defined by $$\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}$$ Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]

The transformation T is defined by the matrix M. The transformation S is defined by the matrix \(\mathbf{M}^{-1}\). Given that the point \((x, y)\) is invariant under transformation T, prove that \((x, y)\) is also an invariant point under transformation S. [3 marks]

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]

You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T.

1. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of \(k\). [4]
2. Determine whether the line with equation \(y = kx\) in part (a) is a line of invariant points under T. [1]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]

You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).

1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]

It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf{M}^n\), for any positive integer \(n\).

1. Explain geometrically why this claim is true. [2]
2. Verify algebraically that this claim is true. [3]

Transformation M is represented by matrix \(\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\).

1. On the diagram in the Printed Answer Booklet draw the image of the unit square under M. [2]
2. 1. Show that there is a constant \(k\) such that \(\mathbf{M} \begin{pmatrix} x \\ kx \end{pmatrix} = 5 \begin{pmatrix} x \\ kx \end{pmatrix}\) for all \(x\). [2]
   2. Hence find the equation of an invariant line under M. [1]
   3. Draw the invariant line from part (ii) (B) on your diagram for part (i). [1]

The transformation \(T\) in the plane consists of a translation in which the point \((x, y)\) is transformed to the point \((x - 1, y + 1)\), followed by a reflection in the line \(y = x\).

1. Determine the \(3 \times 3\) matrix which represents \(T\). [4]
2. Find the equation of the line of fixed points of \(T\). [2]
3. Find \(T^2\) and hence write down \(T^{-1}\). [3]

The transformation \(T\) in the plane consists of a reflection in the line \(y = x\), followed by a translation in which the point \((x, y)\) is transformed to the point \((x + 1, y - 2)\), followed by an anticlockwise rotation through \(90°\) about the origin.

1. Find the \(3 \times 3\) matrix representing \(T\). [6]
2. Show that \(T\) has no fixed points. [3]

The \(2 \times 2\) matrix M is defined by $$\mathbf{M} = \begin{pmatrix} 0 & 0.25 \\ 0.36 & 0 \end{pmatrix}$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by M. [4]

$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ where \(a\) and \(b\) are non-zero constants. Given that the matrix \(\mathbf{A}\) is self-inverse,

1. determine the value of \(b\) and the possible values for \(a\). [5] The matrix \(\mathbf{A}\) represents a linear transformation \(M\). Using the smaller value of \(a\) from part (a),
2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line. [3]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\) [4]

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]

\(\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}\), where \(k\) is a constant.

1. Show that the matrix \(\mathbf{A}\) is non-singular for all values of \(k\). [2]

A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\). The point \(P\) has position vector \(\begin{pmatrix} a \\ 2a \end{pmatrix}\) relative to an origin \(O\). The point \(Q\) has position vector \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\) relative to \(O\). Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),

1. determine the value of \(a\) and the value of \(k\). [3]

Given that, for a different value of \(k\), \(T\) maps the line \(y = 2x\) onto itself,

1. determine this value of \(k\). [3]

A linear transformation T of the \(x\)-\(y\) plane has an associated matrix M, where \(\mathbf{M} = \begin{pmatrix} \lambda & k \\ 1 & \lambda - k \end{pmatrix}\), and \(\lambda\) and \(k\) are real constants.

1. You are given that \(\det \mathbf{M} > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\). [3]
   2. What is the significance of the condition \(\det \mathbf{M} > 0\) for the transformation T? [1]

For the remainder of this question, take \(k = -2\).

1. Determine whether there are any lines through the origin that are invariant lines for the transformation T. [4]

\(\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}\) The matrix \(\mathbf{A}\) represents the linear transformation \(M\). Prove that, for the linear transformation \(M\), there are no invariant lines. [5]

$$\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}$$ The matrix \(\mathbf{A}\) represents the linear transformation \(M\). Prove that, for the linear transformation \(M\), there are no invariant lines. [5]

$$\mathbf{M} = \begin{pmatrix} -2 & 5 \\ 6 & k \end{pmatrix}$$ where \(k\) is a constant. Given that $$\mathbf{M}^2 + 11\mathbf{M} = a\mathbf{I}$$ where \(a\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix,

1. 1. determine the value of \(a\)
   2. show that \(k = -9\) [3]
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf{M}\). [6]
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer. [1]

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]

Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]

The matrix **M** is given by \(\mathbf{M} = \begin{pmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{pmatrix}\).

1. The diagram in the Printed Answer Booklet shows the unit square \(OABC\). The image of the unit square under the transformation represented by **M** is \(OA'B'C'\). Draw and clearly label \(OA'B'C'\). [3]
2. Find the equation of the line of invariant points of this transformation. [3]
3. 1. Find the determinant of **M**. [1]
   2. Describe briefly how this value relates to the transformation represented by **M**. [2]