3.6 Matrix Transformations

The diagram below shows a triangle .

Write down the position matrix of the triangle ABC.

The triangle  is to be mapped to triangle  by a single transformation defined by the transformation matrix 

Find the position matrix of the mapped image and draw triangle on the diagram.

Describe fully the transformation that triangle ABC has undergone.

Points in a plane are subjected to a transformation  that transforms a point  to the point , where  is defined by

Describe in words the transformation T.

Find the matrix .

Hence find the coordinates of the point if .

A quadrilateral has vertices  and .

Find the area of the quadrilateral.

The quadrilateral undergoes a transformation represented by the matrix 

Find the determinant of the transformation matrix.

Hence find the area of the image.

An object undergoes a vertical stretch with scale factor 2 followed by a reflection in the -axis. The position matrix of the image is .

Use a matrix method to find the coordinates of the object before the transformation.

The points and  are transformed to become the points  and  respectively.

Find the matrix that represents the linear transformation.

Given that point is transformed by , find the coordinates of the image point .

An object is reflected in the line .

Write down the matrix that represents the transformation.

 is a vertex of the object that is being reflected.

Find the coordinates of if the coordinates of its image  are .

The triangle , with vertices  and  is translated by the vector  and then enlarged by a scale factor of 3 with the centre of enlargement at the origin.

Find a single transformation in the form that maps  onto .

Hence determine the coordinates of .

Find the transformation matrices that represent the following transformations:

(i)

, a rotation of radians anti-clockwise

(ii)

, a reflection in the line 

(iii)

, a stretch with scale factor 5 parallel to the -axis.

Find a single transformation matrix that represents the composite transformation 

(i)

 

(ii)

Find the coordinates of the image of the point after it has undergone the composite transformation specified in part (b)(i).

State the name of the single transformation that is equivalent to the composite transformation specified in part (b)(ii).

The triangle  with position matrix  has vertices  and .  

The triangle is transformed by the transformation matrix  .  

  denotes the position matrix of the image triangle after  has been transformed  times by matrix .

By multiplying two appropriate single transformation matrices together, verify that the matrix is an enlargement by scale factor followed by a  clockwise rotation.

Explain why the area of the triangle with position matrix will be  of the area of triangle .   

Find and hence the coordinates of the image triangle after triangle  is transformed twice by matrix .

A trapezoid is shown in the diagram below.

By first transforming the trapezoid so that the base is parallel to the y-axis, calculate the area of the trapezoid.

Points in a plane are subject to a transformation  that transforms a point  to the point , where  and are defined by

The position matrix of a series of transformed points is  

Given that the final position matrix of the points is a reflection in the y-axis from their original position, find

A triangle  undergoes a transformation represented by the matrix after which it has an area of 48 cm². 

The original triangle  is then transformed by , where is defined as a stretch with scale factor 3 parallel to the x-axis and a stretch with scale factor parallel to the y-axis. 

Given that the area of triangle  after being transformed by is 84 cm², find .

Consider the functions  and  defined by  and . 

Let the transformation of a point be represented by . 

Use a matrix method to determine the coordinates of a point after undergoing the transformation , given that .

 is a  matrix  that represents the transformation of points  and to  points and respectively.

Given that point  is transformed by , find the coordinates of the image point .

Show that a transformation matrix representing a reflection is a self-inverting matrix

Consider the general complex number  with position vector  and a second complex number .

State the two transformations that occur when is multiplied by .

Hence write down the single matrix, that represents the two transformations from part (a).

Given that and and using the result from part (b), find the position vector of the result of .

By multiplying the two numbers together in their complex form, verify that your answer to part (c) is correct.

The matrices and  are defined by , a stretch with scale factor 4 and y-axis invariant, , a reflection in the line  and . 

Triangle  is mapped onto triangle  by the transformation represented by .

Given triangle  is a rotation of triangle  by   rad clockwise about the origin, find .

The triangle  with position matrix  has vertices ,  and  

The triangle is transformed by a matrix M comprising a counter-clockwise rotation of    about the point .

Given that denotes the position matrix of the image triangle after  has been transformed  times by matrix , find .

The same original triangle  is transformed by a matrix  comprising a clockwise rotation of    about the origin followed by a translation of . 

denotes the position matrix of the image triangle after  has been transformed times by matrix .

Find the distance between the vertices corresponding to the initial point of the triangle with position matrix  and the triangle with position matrix .