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URL: https://nakafa.com/en/subjects/mathematics/geometric-transformation/reflection-matrix
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/geometric-transformation/reflection-matrix/en.mdx

Learn reflection matrices for coordinate axes and diagonal lines. Learn to transform points and triangles with detailed matrix calculations.

---

## Reflection Matrix for Horizontal Axis

Recall that reflecting a point $$(x,y)$$ across the $$x$$-axis results in the image $$(x,-y)$$.

Visible text: Recall that reflecting a point across the -axis results in the image .

We are looking for a matrix $$\begin{pmatrix} r & s \\ t & u \end{pmatrix}$$ such that:

Visible text: We are looking for a matrix such that:

Component: MathContainer
Children:

```math
\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ -y \end{pmatrix}
```

```math
\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} 1x + 0y \\ 0x - 1y \end{pmatrix}
```

By equating the coefficients, we get:

- $$rx + sy = 1x + 0y \implies r=1, s=0$$
- $$tx + uy = 0x - 1y \implies t=0, u=-1$$

Visible text: - 
-

Thus, the reflection matrix for the $$x$$-axis is $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$.

Visible text: Thus, the reflection matrix for the -axis is .

## Reflection Matrix for Vertical Axis

Reflecting a point $$(x,y)$$ across the $$y$$-axis results in $$(-x,y)$$.

Visible text: Reflecting a point across the -axis results in .

Component: MathContainer
Children:

```math
\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -x \\ y \end{pmatrix}
```

```math
\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} -1x + 0y \\ 0x + 1y \end{pmatrix}
```

This gives $$r=-1, s=0, t=0, u=1$$.

Visible text: This gives .

The matrix is $$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$$.

Visible text: The matrix is .

## Reflection Matrix for Main Diagonal Line

Reflecting a point $$(x,y)$$ across the line $$y=x$$ results in $$(y,x)$$.

Visible text: Reflecting a point across the line results in .

Component: MathContainer
Children:

```math
\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} y \\ x \end{pmatrix}
```

```math
\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} 0x + 1y \\ 1x + 0y \end{pmatrix}
```

This gives $$r=0, s=1, t=1, u=0$$.

Visible text: This gives .

The matrix is $$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$.

Visible text: The matrix is .

## Reflection Matrix for Negative Diagonal Line

Reflecting a point $$(x,y)$$ across the line $$y=-x$$ results in $$(-y,-x)$$.

Visible text: Reflecting a point across the line results in .

Component: MathContainer
Children:

```math
\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -y \\ -x \end{pmatrix}
```

```math
\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} 0x - 1y \\ -1x + 0y \end{pmatrix}
```

This gives $$r=0, s=-1, t=-1, u=0$$.

Visible text: This gives .

The matrix is $$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$.

Visible text: The matrix is .

## Basic Reflection Matrices

- Reflection matrix for the $$x$$-axis: $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
- Reflection matrix for the $$y$$-axis: $$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$$
- Reflection matrix for line $$y=x$$: $$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$
- Reflection matrix for line $$y=-x$$: $$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

Visible text: - Reflection matrix for the -axis: 
- Reflection matrix for the -axis: 
- Reflection matrix for line : 
- Reflection matrix for line :

## Finding the Image of a Point using Matrix

Find the image of $$(3,-4)$$ reflected across the $$x$$-axis.

Visible text: Find the image of reflected across the -axis.

**Alternative Solution:**

Using the reflection matrix for the $$x$$-axis:

Visible text: Using the reflection matrix for the -axis:

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 3 \\ -4 \end{pmatrix} = \begin{pmatrix} (1)(3) + (0)(-4) \\ (0)(3) + (-1)(-4) \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}
```

Thus, the image is $$(3,4)$$.

Visible text: Thus, the image is .

## Finding the Image of a Triangle using Matrix

Determine the image of triangle $$ABC$$ with vertices $$A(3,1)$$, $$B(-2,3)$$, and $$C(2,-1)$$ reflected across the $$y$$-axis!

Visible text: Determine the image of triangle with vertices , , and reflected across the -axis!

**Alternative Solution:**

The reflection matrix for the $$y$$-axis is $$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$$.

Visible text: The reflection matrix for the -axis is .

The matrix of triangle $$ABC$$'s vertices: $$\begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}$$.

Visible text: The matrix of triangle 's vertices: .

Component: MathContainer
Children:

```math
\begin{pmatrix} x'_A & x'_B & x'_C \\ y'_A & y'_B & y'_C \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}
```

```math
= \begin{pmatrix} (-1)(3)+0 & (-1)(-2)+0 & (-1)(2)+0 \\ 0+(1)(1) & 0+(1)(3) & 0+(1)(-1) \end{pmatrix} = \begin{pmatrix} -3 & 2 & -2 \\ 1 & 3 & -1 \end{pmatrix}
```

Thus, the image is triangle <InlineMath math="A'B'C'" /> with vertices <InlineMath math="A'(-3,1)" />, <InlineMath math="B'(2,3)" />, and <InlineMath math="C'(-2,-1)" />.

Component: LineEquation
Props:
- title: Reflection of $$\triangle ABC$$ over $$y$$-axis using Matrix
  Visible text: Reflection of over -axis using Matrix
- description: Triangle $$ABC$$ is reflected to become{" "}
<InlineMath math="A'B'C'" /> over the $$y$$-axis.
  Visible text: Triangle is reflected to become{" "}
<InlineMath math="A'B'C'" /> over the -axis.
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},
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from: { x: -2, y: 3, z: 0, label: "B(-2,3)" },
to: { x: 2, y: -1, z: 0, label: "C(2,-1)" },
},
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from: { x: 2, y: -1, z: 0, label: "C(2,-1)" },
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to: { x: 2, y: 3, z: 0, label: "B'(2,3)" },
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from: { x: 2, y: 3, z: 0, label: "B'(2,3)" },
to: { x: -2, y: -1, z: 0, label: "C'(-2,-1)" },
},
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from: { x: -2, y: -1, z: 0, label: "C'(-2,-1)" },
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## Exercises

1.  Find the image of point $$(3,5)$$ reflected across the $$x$$-axis using the matrix multiplication $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$.
2.  Determine the image of triangle $$ABC$$ with vertices $$A(3,1)$$, $$B(-2,3)$$, and $$C(2,-1)$$ reflected across the line $$y=-x$$!

Visible text: 1. Find the image of point reflected across the -axis using the matrix multiplication .
2. Determine the image of triangle with vertices , , and reflected across the line !

### Key Answers

1.  The image of point $$(3,5)$$ reflected across the $$x$$-axis using the matrix multiplication $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ is:

    <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 3 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}" />

    Image: $$(3,-5)$$.

2.  The reflection matrix for $$y=-x$$ is $$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$.

    Matrix of ABC vertices: $$\begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}$$.

    <MathContainer>
      
    
    ```math
    \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}
    ```

      
    
    ```math
    = \begin{pmatrix} (0)(3)+(-1)(1) & (0)(-2)+(-1)(3) & (0)(2)+(-1)(-1) \\ (-1)(3)+(0)(1) & (-1)(-2)+(0)(3) & (-1)(2)+(0)(-1) \end{pmatrix}
    ```

      
    
    ```math
    = \begin{pmatrix} -1 & -3 & 1 \\ -3 & 2 & -2 \end{pmatrix}
    ```

    </MathContainer>

    Image: <InlineMath math="A'(-1,-3)" />, <InlineMath math="B'(-3,2)" />, <InlineMath math="C'(1,-2)" />.

Visible text: 1. The image of point reflected across the -axis using the matrix multiplication is:

 <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 3 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}" />

 Image: .

2. The reflection matrix for is .

 Matrix of ABC vertices: .

 <MathContainer>
 
 

 
 

 
 

 </MathContainer>

 Image: <InlineMath math="A'(-1,-3)" />, <InlineMath math="B'(-3,2)" />, <InlineMath math="C'(1,-2)" />.