# Reflection in the Origin Matrix

How to use the reflection in the origin matrix to find the image under the reflection: the matrix, example, and its solution.

## Matrix

The reflection in the origin matrix is
[-1 0 / 0 -1],
which is -I.

(I is the identity matrix.)

To find the coordinates of the image,
multiply the reflection matrix [-1 0 / 0 -1]
in front of the vertex matrix.

Reflection in the origin

Multiplying matrices

## Example

Previously, you've solved this example.

Reflection in the origin

Let's solve the same example
by using the reflection in the origin matrix.

A(-3, 5), B(5, 4), C(2, -1)

So the vertex matrix is
[-3 5 2 / 5 4 -1].

The reflection in the origin matrix is
[-1 0 / 0 -1].

So multiply the reflection matrix [-1 0 / 0 -1]
in front of the vertex matrix [-3 5 2 / 5 4 -1].

Multiply these two matrices.

Multiplying matrices

[Row 1]⋅[Column 1]: (-1)⋅(-3) + 0⋅5
[Row 1]⋅[Column 2]: (-1)⋅5 + 0⋅4
[Row 1]⋅[Column 3]: (-1)⋅2 + 0⋅(-1)

[Row 2]⋅[Column 1]: 0⋅(-3) + (-1)⋅5
[Row 2]⋅[Column 2]: 0⋅5 + (-1)⋅4
[Row 2]⋅[Column 3]: 0⋅2 + (-1)⋅(-1)

Then the vertex matrix of the image is
[3 -5 -2 / -5 -4 1].

The first column is [3 / -5].
So A'(3, -5).

The second column is [-5 / -4].
So B'(-5, -4).

The third column is [-2 / 1].
So C'(-2, 1).

These are the coordinates of the image.

Here's what you've solved.

By multiplying
the reflection in the origin matrix [-1 0 / 0 -1],
you found the vertices of the image (△A'B'C')
under the reflection in the origin.