Math Problem Statement

Analyze the mathematical questions in the uploaded image and provide solutions.

Solution

The uploaded image contains three mathematical questions related to transformations, matrix operations, and rotation in geometry. I will provide a detailed explanation and solutions for each question.

1. True or False Statements

  1. Sebuah titik yang ditransformasikan pasti berpindah kuadran.

    • Solution: Not all transformations cause a point to change quadrants. For example, a reflection across an axis or a rotation of 360° might leave a point in the same quadrant. Answer: Salah (False).

  2. Suatu titik di kuadran II jika dicerminkan terhadap sumbu x bayangannya ada di kuadran II.

    • Solution: When a point in Quadrant II is reflected over the x-axis, it moves to Quadrant IV, not Quadrant II. Answer: Salah (False).

  3. Suatu titik yang ada di kuadran I jika dirotasikan 360° bayangannya adalah titik itu sendiri.

    • Solution: A 360° rotation returns a point to its original position regardless of the quadrant. Answer: Benar (True).

2. Matrix Operation Question

The matrix operation involves the transformation:

k & 0 \\ 0 & k \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$ - **Solution**: Performing the matrix multiplication: $$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} kx \\ ky \end{bmatrix}$$ This is equivalent to multiplying the vector $$\begin{bmatrix} x \\ y \end{bmatrix}$$ by the scalar $$k$$. **Answer**: **A. Benar** (True). --- ### **3. Rotation Transformation** - The question asks about the transformation matrix for a 90° rotation about the origin, specifically $$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$. - **Solution**: A 90° counterclockwise rotation is represented by the matrix: $$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$ This matrix is correct for a 90° rotation about the origin. **Answer**: The given matrix is correct. --- ### Summary of Answers: 1. (1) Salah, (2) Salah, (3) Benar. 2. A. Benar. 3. The given matrix is correct. --- Would you like more detailed explanations or derivations for these solutions? Here are some related questions you might find useful: 1. What is the general formula for a reflection matrix? 2. How do you calculate a rotation matrix for angles other than 90°? 3. Can you explain how to identify a transformation's effect on a point's quadrant? 4. What other examples of scalar multiplication matrices are there? 5. How does the determinant of a rotation matrix indicate its validity? **Tip**: Always sketch the transformation or apply a test point to verify its behavior when unsure.

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Math Problem Analysis

Mathematical Concepts

Geometry transformations
Matrix operations
Rotation matrices

Formulas

Reflection transformation: Matrix representation depends on the axis of reflection.
Rotation transformation matrix for θ = 90°: [[0, 1], [-1, 0]]
Matrix scalar multiplication formula: k * [[x], [y]] = [[kx], [ky]]

Theorems

Properties of rotational symmetry
Basic properties of matrix transformations

Suitable Grade Level

Grades 9-12

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