Reflection Matrix: y = x

How to use the reflection in y = x matrix to find the image under the reflection: formula, 1 example, and its solution.

Formula

Formula

The reflection in y = x matrix is
[0 1 / 1 0].

Example

Example

Solution

The image is under
the reflection in y = x.

So write the reflection in y = x matrix
[0 1 / 1 0].

Write the vertex matrix.

A(6, 4), B(7, 1), C(2, -2)

So the vertex matrix is
[6 7 2 / 4 1 -2].

So the vertex matrix of the image is
[0 1 / 1 0][6 7 2 / 4 1 -2].

Solve [0 1 / 1 0][6 7 2 / 4 1 -2].

Multiply Matrices

Row 1, column 1:
0⋅6 + 1⋅4

Row 1, column 2:
0⋅7 + 1⋅1

Row 1, column 3:
0⋅2 + 1⋅(-2)

Row 2, column 1:
1⋅6 + 0⋅4

Row 2, column 2:
1⋅7 + 0⋅1

Row 2, column 3:
1⋅2 + 0⋅(-2)

This is the vertex matrix of the image.

0⋅6 + 1⋅4
= 0 + 4

0⋅7 + 1⋅1
= 0 + 1

0⋅2 + 1⋅(-2)
= 0 - 2

1⋅6 + 0⋅4
= 6 + 0

1⋅7 + 0⋅1
= 7 + 0

1⋅2 + 0⋅(-2)
= 2 + 0

0 + 4 = 4

0 + 1 = 1

0 - 2 = -2

6 + 0 = 6

7 + 0 = 7

2 + 0 = 2

[4 1 -2 / 6 7 2]
is the vertex matrix of the image.

So column 1 is the image of A:
A'(4, 6).

Column 2 is the image of B:
B'(1, 7).

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

So
A'(4, 6)
B'(1, 7)
C'(-2, 2)
is the answer.

Graph

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

The image is under
the reflection in y = x.