Dilation Matrix

How to use the dilation matrix to find the image under a dilation: formula, 1 example, and its solution.

Formula

Formula

For the dilation of k,
the dilation matrix is
kI = [k 0 / 0 k].

I: Identity matrix

Example

Example

Solution

The image is under
the dilation of 2.

So write the dilation matrix
2I.

Write the vertex matrix.

A(-2, 1), B(2, 3), C(0, -1)

So the vertex matrix is
[-2 2 0 / 1 3 -1].

So the vertex matrix of the image is
2I[-2 2 0 / 1 3 -1].

I is the identity matrix.

So, by its definition,
I[-2 2 0 / 1 3 -1]
= [-2 2 0 / 1 3 -1].

Multiply 2 to each element.

[-4 4 0 / 2 6 -2]
is the vertex matrix of the image.

So column 1 is the image of A:
A'(-4, 2).

Column 2 is the image of B:
B'(4, 6).

Column 3 is the image of C:
C'(0, -2).

So
A'(-4, 2)
B'(4, 6)
C'(0, -2)
is the answer.

Graph

This is the graph of △ABC
and its image △A'B'C'.

The image is under the dilation of 2.