# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/11/mathematics/geometric-transformation/reflection-matrix
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/geometric-transformation/reflection-matrix/en.mdx

Output docs content for large language models.

---

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

export const metadata = {
  title: "Reflection Matrix",
  description: "Master reflection matrices for X-axis, Y-axis, y=x, and y=-x lines. Learn to transform points and triangles with detailed matrix calculations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/10/2025",
  subject: "Geometric Transformation",
};

## Reflection Matrix for X-axis

Recall that reflecting a point <InlineMath math="(x,y)" /> across the X-axis results in the image <InlineMath math="(x,-y)" />.

We are looking for a matrix <InlineMath math="\begin{pmatrix} r & s \\ t & u \end{pmatrix}" /> such that:

<div className="flex flex-col gap-4">
  <BlockMath math="\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ -y \end{pmatrix}" />
  <BlockMath math="\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} 1x + 0y \\ 0x - 1y \end{pmatrix}" />
</div>

By equating the coefficients, we get:

- <InlineMath math="rx + sy = 1x + 0y \implies r=1, s=0" />
- <InlineMath math="tx + uy = 0x - 1y \implies t=0, u=-1" />

Thus, the reflection matrix for the X-axis is <InlineMath math="\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}" />.

## Reflection Matrix for Y-axis

Reflecting a point <InlineMath math="(x,y)" /> across the Y-axis results in <InlineMath math="(-x,y)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -x \\ y \end{pmatrix}" />
  <BlockMath math="\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} -1x + 0y \\ 0x + 1y \end{pmatrix}" />
</div>

This gives <InlineMath math="r=-1, s=0, t=0, u=1" />.

The matrix is <InlineMath math="\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}" />.

## Reflection Matrix for Line y = x

Reflecting a point <InlineMath math="(x,y)" /> across the line <InlineMath math="y=x" /> results in <InlineMath math="(y,x)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} y \\ x \end{pmatrix}" />
  <BlockMath math="\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} 0x + 1y \\ 1x + 0y \end{pmatrix}" />
</div>

This gives <InlineMath math="r=0, s=1, t=1, u=0" />.

The matrix is <InlineMath math="\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}" />.

## Reflection Matrix for Line y = -x

Reflecting a point <InlineMath math="(x,y)" /> across the line <InlineMath math="y=-x" /> results in <InlineMath math="(-y,-x)" />.

<div className="flex flex-col gap-4">
  <BlockMath math="\begin{pmatrix} r & s \\ t & u \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -y \\ -x \end{pmatrix}" />
  <BlockMath math="\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} 0x - 1y \\ -1x + 0y \end{pmatrix}" />
</div>

This gives <InlineMath math="r=0, s=-1, t=-1, u=0" />.

The matrix is <InlineMath math="\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}" />.

## Basic Reflection Matrices

- Reflection matrix for X-axis: <InlineMath math="\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}" />
- Reflection matrix for Y-axis: <InlineMath math="\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}" />
- Reflection matrix for line <InlineMath math="y=x" />: <InlineMath math="\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}" />
- Reflection matrix for line <InlineMath math="y=-x" />: <InlineMath math="\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}" />

## Finding the Image of a Point using Matrix

Find the image of <InlineMath math="(3,-4)" /> reflected across the X-axis.

**Alternative Solution:**

Using the reflection matrix for the X-axis:

<BlockMath 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 <InlineMath math="(3,4)" />.

## Finding the Image of a Triangle using Matrix

Determine the image of triangle ABC with vertices <InlineMath math="A(3,1)" />, <InlineMath math="B(-2,3)" />, and <InlineMath math="C(2,-1)" /> reflected across the Y-axis!

**Alternative Solution:**

The reflection matrix for the Y-axis is <InlineMath math="\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}" />.

The matrix of the triangle's vertices ABC: <InlineMath math="\begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}" />.

<div className="flex flex-col gap-4">
  <BlockMath 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}" />
  <BlockMath 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}" />
</div>

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)" />.

<LineEquation
  title={
    <>
      Reflection of <InlineMath math="\triangle ABC" /> over Y-axis using Matrix
    </>
  }
  description={
    <>
      Triangle <InlineMath math="ABC" /> is reflected to become{" "}
      <InlineMath math="A'B'C'" /> over the Y-axis.
    </>
  }
  data={[
    ...[
      {
        from: { x: 3, y: 1, z: 0, label: "A(3,1)" },
        to: { x: -2, y: 3, z: 0, label: "B(-2,3)" },
      },
      {
        from: { x: -2, y: 3, z: 0, label: "B(-2,3)" },
        to: { x: 2, y: -1, z: 0, label: "C(2,-1)" },
      },
      {
        from: { x: 2, y: -1, z: 0, label: "C(2,-1)" },
        to: { x: 3, y: 1, z: 0, label: "A(3,1)" },
      },
    ].map((segment) => ({
      points: [segment.from, segment.to],
      color: getColor("AMBER"),
      showPoints: true,
      labels: [{ text: segment.from.label, at: 0, offset: [0.3, 0.3, 0] }],
    })),
    ...[
      {
        from: { x: -3, y: 1, z: 0, label: "A'(-3,1)" },
        to: { x: 2, y: 3, z: 0, label: "B'(2,3)" },
      },
      {
        from: { x: 2, y: 3, z: 0, label: "B'(2,3)" },
        to: { x: -2, y: -1, z: 0, label: "C'(-2,-1)" },
      },
      {
        from: { x: -2, y: -1, z: 0, label: "C'(-2,-1)" },
        to: { x: -3, y: 1, z: 0, label: "A'(-3,1)" },
      },
    ].map((segment) => ({
      points: [segment.from, segment.to],
      color: getColor("TEAL"),
      showPoints: true,
      labels: [{ text: segment.from.label, at: 0, offset: [0.3, 0.3, 0] }],
    })),
  ]}
  showZAxis={false}
  cameraPosition={[0, 0, 12]}
/>

## Exercises

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

### Key Answers

1.  The image of point <InlineMath math="(3,5)" /> reflected across the X-axis using the matrix multiplication <InlineMath math="\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: <InlineMath math="(3,-5)" />.

2.  The reflection matrix for <InlineMath math="y=-x" /> is <InlineMath math="\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}" />.

    Matrix of ABC vertices: <InlineMath math="\begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 3 & -2 & 2 \\ 1 & 3 & -1 \end{pmatrix}" />
      <BlockMath 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}" />
      <BlockMath math="= \begin{pmatrix} -1 & -3 & 1 \\ -3 & 2 & -2 \end{pmatrix}" />
    </div>

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