Matrix transformations

The matrix C is defined by

Prove that the transformation represented by C has no invariant lines of the form y = kx

Which one of the points below is an invariant point under this transformation?

Circle your answer.

(1, 1)
(0, 2)
(3, 0)
(2, 1)

The matrix M is defined as

where c is a real number.

The linear transformation T is represented by the matrix M

Show that, for one particular value of c, the image under T of every point lies in the plane

State the value of c for which this occurs.

It is given that M is a non-singular matrix.

State any restrictions on the value of c

Using your answer from part (b)(ii), solve

Roberto is solving this mathematics problem:

Roberto’s solution is as follows:

Sketch the curve C₁

Explain what Roberto has done wrong.

Find the area enclosed by C₁

P and Q are distinct points on C₁ for which r is a maximum. P is above the initial line.

Find the polar coordinates of P and Q

T maps C₁ onto a curve C₂

T maps P onto the point P

Find the polar coordinates of P

Find the area enclosed by C₂

Fully justify your answer.

The matrix A represents a reflection in the line y mx, where m is a constant.

You may use the result in the formulae booklet.

where is the 2 2 identity matrix and k is an integer.

The diagram below shows a point P and the line y mx

Draw four lines on the diagram to demonstrate the result proved in part (b).

Label as P the image of P under the transformation represented by (BA)²

Explain how your completed diagram shows the result proved in part (b).

Find the value of m such that C BA

Fully justify your answer.

Which one of the matrices below represents a rotation of 90 about the x-axis?

Circle your answer.

Three non-singular square matrices, A, B and R are such that

The matrix R represents a rotation about the z-axis through an angle and

Show that A is independent of the value of .

Give a full description of the single transformation represented by the matrix A.