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

Learn how to represent dilation with matrices and scale shapes from the origin or an arbitrary point.

---

## Finding the Matrix Associated with Dilation

How to find the matrix associated with a dilation operation? Recall that a point $$(x,y)$$ is mapped by a dilation with a factor $$k \neq 0$$ and center $$O$$ to $$(kx,ky)$$.

Visible text: How to find the matrix associated with a dilation operation? Recall that a point is mapped by a dilation with a factor and center to .

Suppose the matrix we are looking for is $$\begin{pmatrix} r & s \\ t & u \end{pmatrix}$$.

Visible text: Suppose the matrix we are looking for is .

Find $$r, s, t, u$$ such that it satisfies

Visible text: Find such that it satisfies

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

From the matrix multiplication on the left side, we get:

```math
\begin{pmatrix} rx + sy \\ tx + uy \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}
```

By equating the corresponding components:

- **First row:** $$rx + sy = kx$$. For this equation to hold for all $$x$$ and $$y$$, the coefficients of $$x$$ must be equal, and the coefficients of $$y$$ must be equal. Thus, $$r = k$$ and $$s = 0$$.
- **Second row:** $$tx + uy = ky$$. Similarly, $$t = 0$$ and $$u = k$$.

Visible text: - **First row:** . For this equation to hold for all and , the coefficients of must be equal, and the coefficients of must be equal. Thus, and .
- **Second row:** . Similarly, and .

## Dilation Matrix with Respect to the Origin

The matrix associated with a dilation by a factor $$k \neq 0$$ with respect to the origin $$O(0,0)$$ is

Visible text: The matrix associated with a dilation by a factor with respect to the origin is

```math
\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}
```

## Matrix Operation for Dilation with Respect to an Arbitrary Point

A point $$(x,y)$$ dilated by a factor $$k \neq 0$$ and center $$(a,b)$$ will be mapped to $$(k(x-a)+a,\, k(y-b)+b)$$.

Visible text: A point dilated by a factor and center will be mapped to .

Find the combination of matrix operations on the position vector $$\begin{pmatrix} x-a \\ y-b \end{pmatrix}$$ such that the result is $$\begin{pmatrix} k(x-a)+a \\ k(y-b)+b \end{pmatrix}$$.

Visible text: Find the combination of matrix operations on the position vector such that the result is .

The matrix operation associated with a dilation by a factor $$k \neq 0$$ with respect to the point $$(a,b)$$ is

Visible text: The matrix operation associated with a dilation by a factor with respect to the point is

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \begin{pmatrix} x-a \\ y-b \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}
```

or it can also be written as:

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} - \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \left( \begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} a \\ b \end{pmatrix} \right)
```

## Finding the Image of a Dilation Using Matrices

Determine the image of point $$A(2,4)$$ transformed by a dilation with a factor of $$2$$ with respect to the center point $$P(1,1)$$!

Visible text: Determine the image of point transformed by a dilation with a factor of with respect to the center point !

**Alternative Solution:**

Given $$x=2, y=4, k=2, a=1, b=1$$.

Visible text: Given .

Component: MathContainer
Children:

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 2-1 \\ 4-1 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}
```

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}
```

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} (2)(1) + (0)(3) \\ (0)(1) + (2)(3) \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}
```

```math
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}
```

Thus, the image of point $$A(2,4)$$ is <InlineMath math="A'(3,7)" />.

Visible text: Thus, the image of point is <InlineMath math="A'(3,7)" />.

Component: LineEquation
Props:
- title: Visualization of Dilation of Point $$A(2,4)$$ with
Center $$P(1,1)$$ and Scale Factor{" "}
$$k=2$$
  Visible text: Visualization of Dilation of Point with
Center and Scale Factor{" "}
- description: Point $$A(2,4)$$ is dilated with respect to the center{" "}
$$P(1,1)$$ with a scale factor $$k=2$${" "}
to produce the image <InlineMath math="A'(3,7)" />. The line from the
center to the original point and from the center to the image lie on the
same line, and the distance <InlineMath math="PA'" /> is twice the
distance $$PA$$.
  Visible text: Point is dilated with respect to the center{" "}
 with a scale factor {" "}
to produce the image <InlineMath math="A'(3,7)" />. The line from the
center to the original point and from the center to the image lie on the
same line, and the distance <InlineMath math="PA'" /> is twice the
distance .
- data: [
{
points: [{ x: 1, y: 1, z: 0 }],
color: getColor("ROSE"),
showPoints: true,
labels: [{ text: "P(1,1)", at: 0, offset: [-0.5, -0.5, 0] }],
},
{
points: [{ x: 2, y: 4, z: 0 }],
color: getColor("SKY"),
showPoints: true,
labels: [{ text: "A(2,4)", at: 0, offset: [1.5, 0.2, 0] }],
},
{
points: [{ x: 3, y: 7, z: 0 }],
color: getColor("EMERALD"),
showPoints: true,
labels: [{ text: "A'(3,7)", at: 0, offset: [0.5, 0.5, 0] }],
},
{
points: [
{ x: 1, y: 1, z: 0 }, // P
{ x: 2, y: 4, z: 0 }, // A
],
color: getColor("AMBER"),
},
{
points: [
{ x: 1, y: 1, z: 0 }, // P
{ x: 3, y: 7, z: 0 }, // A'
],
color: getColor("SKY"),
cone: { position: "end", size: 0.3 },
},
]
- showZAxis: false
- cameraPosition: [0, 0, 18]

## Exercises

1.  Find the coordinates of the image of the point $$(a,b)$$ under the dilation $$[O,3]$$!
2.  Determine the matrix corresponding to a dilation with a scale factor of $$-2$$ and centered at $$O(0,0)$$.
3.  A point $$B(-1, 5)$$ is dilated with center $$P(2, -3)$$ and scale factor $$k = \frac{1}{2}$$. Determine the coordinates of the image of point $$B$$!
4.  A triangle $$KLM$$ with vertices $$K(1,1)$$, $$L(5,1)$$, and $$M(3,4)$$ is dilated with center $$O(0,0)$$ and scale factor $$2$$. Draw the original triangle and its image, then determine the coordinates of the image vertices!

Visible text: 1. Find the coordinates of the image of the point under the dilation !
2. Determine the matrix corresponding to a dilation with a scale factor of and centered at .
3. A point is dilated with center and scale factor . Determine the coordinates of the image of point !
4. A triangle with vertices , , and is dilated with center and scale factor . Draw the original triangle and its image, then determine the coordinates of the image vertices!

### Key Answers

1. The dilation $$[O,3]$$ means the center of dilation is $$O(0,0)$$ and the scale factor is $$k=3$$.

   Let the point be $$Q(a,b)$$.

   Then $$x=a, y=b, k=3$$.

   <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 3a \\ 3b \end{pmatrix}" />

   Thus, the coordinates of the image of point $$(a,b)$$ are $$(3a, 3b)$$.

2. Scale factor $$k=-2$$, center $$O(0,0)$$.

   The dilation matrix is:

   
   
   ```math
   \begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}
   ```

3. Point $$B(-1, 5)$$, center $$P(2, -3)$$, scale factor $$k = \frac{1}{2}$$.

   $$x=-1, y=5, a=2, b=-3, k=\frac{1}{2}$$.

   <MathContainer>
     <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} -1-2 \\ 5-(-3) \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix}" />
     <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} -3 \\ 8 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix}" />
     <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} (\frac{1}{2})(-3) + (0)(8) \\ (0)(-3) + (\frac{1}{2})(8) \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix}" />
     <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -\frac{3}{2} \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -\frac{3}{2} + \frac{4}{2} \\ 4 - 3 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ 1 \end{pmatrix}" />
   </MathContainer>

   The coordinates of the image of point $$B$$ are <InlineMath math="B'(\frac{1}{2}, 1)" />.

4. Triangle $$KLM$$ with $$K(1,1)$$, $$L(5,1)$$, $$M(3,4)$$.

   Center $$O(0,0)$$, $$k=2$$.

   Image of point $$K(1,1)$$:

   <BlockMath math="\begin{pmatrix} x_K' \\ y_K' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} \implies K'(2,2)" />

   Image of point $$L(5,1)$$:

   <BlockMath math="\begin{pmatrix} x_L' \\ y_L' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 5 \\ 1 \end{pmatrix} = \begin{pmatrix} 10 \\ 2 \end{pmatrix} \implies L'(10,2)" />

   Image of point $$M(3,4)$$:

   <ContentStack>

     <BlockMath math="\begin{pmatrix} x_M' \\ y_M' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 8 \end{pmatrix} \implies M'(6,8)" />

     <LineEquation
       title={
         <>
           Visualization of Dilation of Triangle $$KLM$$ with
           Center $$O(0,0)$$ and Scale Factor{" "}
           $$k=2$$
         </>
       }
       description={
         <>
           Triangle $$KLM$$ is dilated to become triangle{" "}
           <InlineMath math="K'L'M'" />. The center of dilation is{" "}
           $$O(0,0)$$.
         </>
       }
       data={[
         // Original Triangle KLM
         {
           points: [
             { x: 1, y: 1, z: 0 }, // K
             { x: 5, y: 1, z: 0 }, // L
           ],
           color: getColor("SKY"),
           labels: [
             { text: "K(1, 1)", at: 0, offset: [-0.7, -0.2, 0] },
             { text: "L(5, 1)", at: 1, offset: [0.7, -0.2, 0] },
           ],
           showPoints: true,
         },
         {
           points: [
             { x: 5, y: 1, z: 0 }, // L
             { x: 3, y: 4, z: 0 }, // M
           ],
           color: getColor("ORANGE"),
           labels: [{ text: "M(3,4)", at: 1, offset: [0, 0.5, 0] }],
           showPoints: true,
         },
         {
           points: [
             { x: 3, y: 4, z: 0 }, // M
             { x: 1, y: 1, z: 0 }, // K
           ],
           color: getColor("PURPLE"),
           showPoints: true,
         },
         // Dilated Triangle K'L'M'
         {
           points: [
             { x: 2, y: 2, z: 0 }, // K'
             { x: 10, y: 2, z: 0 }, // L'
           ],
           color: getColor("TEAL"),
           labels: [
             { text: "K'(2, 2)", at: 0, offset: [-0.8, -0.3, 0] },
             { text: "L'(10, 2)", at: 1, offset: [0.8, -0.3, 0] },
           ],
           showPoints: true,
         },
         {
           points: [
             { x: 10, y: 2, z: 0 }, // L'
             { x: 6, y: 8, z: 0 }, // M'
           ],
           color: getColor("PINK"),
           labels: [{ text: "M'(6,8)", at: 1, offset: [0, 0.6, 0] }],
           showPoints: true,
         },
         {
           points: [
             { x: 6, y: 8, z: 0 }, // M'
             { x: 2, y: 2, z: 0 }, // K'
           ],
           color: getColor("INDIGO"),
           showPoints: true,
         },
         // Origin
         {
           points: [{ x: 0, y: 0, z: 0 }],
           color: getColor("INDIGO"),
           showPoints: true,
           labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
         },
       ]}
       showZAxis={false}
       cameraPosition={[0, 0, 20]}
     />
   </ContentStack>

Visible text: 1. The dilation means the center of dilation is and the scale factor is .

 Let the point be .

 Then .

 <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 3a \\ 3b \end{pmatrix}" />

 Thus, the coordinates of the image of point are .

2. Scale factor , center .

 The dilation matrix is:

 
 

3. Point , center , scale factor .

 .

 <MathContainer>
 <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} -1-2 \\ 5-(-3) \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix}" />
 <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} -3 \\ 8 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix}" />
 <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} (\frac{1}{2})(-3) + (0)(8) \\ (0)(-3) + (\frac{1}{2})(8) \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix}" />
 <BlockMath math="\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -\frac{3}{2} \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -\frac{3}{2} + \frac{4}{2} \\ 4 - 3 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ 1 \end{pmatrix}" />
 </MathContainer>

 The coordinates of the image of point are <InlineMath math="B'(\frac{1}{2}, 1)" />.

4. Triangle with , , .

 Center , .

 Image of point :

 <BlockMath math="\begin{pmatrix} x_K' \\ y_K' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} \implies K'(2,2)" />

 Image of point :

 <BlockMath math="\begin{pmatrix} x_L' \\ y_L' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 5 \\ 1 \end{pmatrix} = \begin{pmatrix} 10 \\ 2 \end{pmatrix} \implies L'(10,2)" />

 Image of point :

 <ContentStack>

 <BlockMath math="\begin{pmatrix} x_M' \\ y_M' \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 8 \end{pmatrix} \implies M'(6,8)" />

 <LineEquation
 title={
 <>
 Visualization of Dilation of Triangle with
 Center and Scale Factor{" "}
 
 </>
 }
 description={
 <>
 Triangle is dilated to become triangle{" "}
 <InlineMath math="K'L'M'" />. The center of dilation is{" "}
 .
 </>
 }
 data={[
 // Original Triangle KLM
 {
 points: [
 { x: 1, y: 1, z: 0 }, // K
 { x: 5, y: 1, z: 0 }, // L
 ],
 color: getColor("SKY"),
 labels: [
 { text: "K(1, 1)", at: 0, offset: [-0.7, -0.2, 0] },
 { text: "L(5, 1)", at: 1, offset: [0.7, -0.2, 0] },
 ],
 showPoints: true,
 },
 {
 points: [
 { x: 5, y: 1, z: 0 }, // L
 { x: 3, y: 4, z: 0 }, // M
 ],
 color: getColor("ORANGE"),
 labels: [{ text: "M(3,4)", at: 1, offset: [0, 0.5, 0] }],
 showPoints: true,
 },
 {
 points: [
 { x: 3, y: 4, z: 0 }, // M
 { x: 1, y: 1, z: 0 }, // K
 ],
 color: getColor("PURPLE"),
 showPoints: true,
 },
 // Dilated Triangle K'L'M'
 {
 points: [
 { x: 2, y: 2, z: 0 }, // K'
 { x: 10, y: 2, z: 0 }, // L'
 ],
 color: getColor("TEAL"),
 labels: [
 { text: "K'(2, 2)", at: 0, offset: [-0.8, -0.3, 0] },
 { text: "L'(10, 2)", at: 1, offset: [0.8, -0.3, 0] },
 ],
 showPoints: true,
 },
 {
 points: [
 { x: 10, y: 2, z: 0 }, // L'
 { x: 6, y: 8, z: 0 }, // M'
 ],
 color: getColor("PINK"),
 labels: [{ text: "M'(6,8)", at: 1, offset: [0, 0.6, 0] }],
 showPoints: true,
 },
 {
 points: [
 { x: 6, y: 8, z: 0 }, // M'
 { x: 2, y: 2, z: 0 }, // K'
 ],
 color: getColor("INDIGO"),
 showPoints: true,
 },
 // Origin
 {
 points: [{ x: 0, y: 0, z: 0 }],
 color: getColor("INDIGO"),
 showPoints: true,
 labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
 },
 ]}
 showZAxis={false}
 cameraPosition={[0, 0, 20]}
 />
 </ContentStack>