Let’s solve each problem one by one, step by step.
---
Problem 1:
Matrix:
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\]
This is a standard rotation matrix. The general form for counterclockwise rotation by θ is:
\[
\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}
\]
For θ = 90°:
- cos(90°) = 0
- sin(90°) = 1
So the matrix becomes:
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\]
✔ That matches exactly. So this is a
counterclockwise rotation about the origin by 90°.
---
Problem 2:
Matrix:
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
\]
This multiplies both x and y by -1 → (x, y) → (-x, -y)
That’s the same as rotating 180° around the origin.
Check rotation matrix for 180°:
- cos(180°) = -1
- sin(180°) = 0
Rotation matrix:
\[
\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
\]
✔ Matches. So it’s a
rotation about the origin by 180°.
---
Problem 3:
Matrix:
\[
\begin{bmatrix}
4 & 0 \\
0 & 4
\end{bmatrix}
\]
This scales both x and y by 4 → (x, y) → (4x, 4y)
That’s a uniform scaling — called a
dilation with scale factor 4.
✔ Correct answer:
A dilation with scale factor 4
---
Problem 4:
Matrix:
\[
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
Apply to point (x, y):
New point = (0*x + 1*y, 1*x + 0*y) = (y, x)
Swapping x and y reflects over the line
y = x.
Example: (2, 5) → (5, 2) → that’s reflection over y=x.
✔ So correct answer:
A reflection about the line y = x
---
Problem 5:
“A vertical stretch with scale factor 1/6”
Vertical stretch means we multiply the y-coordinate by 1/6, leave x unchanged.
So transformation: (x, y) → (x, (1/6)y)
Matrix form:
\[
\begin{bmatrix}
1 & 0 \\
0 & \frac{1}{6}
\end{bmatrix}
\]
✔ Fill in the boxes:
Top row: 1 , 0
Bottom row: 0 , 1/6
---
Problem 6:
“A vertical stretch with scale factor 6”
Same idea: (x, y) → (x, 6y)
Matrix:
\[
\begin{bmatrix}
1 & 0 \\
0 & 6
\end{bmatrix}
\]
✔ Fill in the boxes:
Top row: 1 , 0
Bottom row: 0 , 6
---
Final Answer:
1. A counterclockwise rotation about the origin by 90°
2. A rotation about the origin by 180°
3. A dilation with scale factor 4
4. A reflection about the line y = x
5. Matrix:
[1 0]
[0 1/6]
6. Matrix:
[1 0]
[0 6]
Parent Tip: Review the logic above to help your child master the concept of transformations with matrices worksheet.
