Matrix Transformation Invariance?

📌 CONCEPT: A matrix transformation that doubles the area of any given rectangle maintains the shape of the rectangle, but scales it up by a factor of √2:1 in both the x and y directions.

📐 RULE / FORMULA: The transformation matrix is A = [2 0] [0 2], which represents a scaling factor of 2 in both the x and y directions.

💡 WORKED EXAMPLE: Consider a rectangle with sides 3 and 4. The area of the original rectangle is 3 × 4 = 12. After transformation, the area becomes 2 × 12 = 24. To find the dimensions of the resulting rectangle, we need to find the new sides that give an area of 24. Since the transformation maintains the shape, the new sides will be √(24) in both the x and y directions. Calculating, we get √(24) = √(4 × 6) = 2√6. Therefore, the dimensions of the resulting rectangle are 2√6 and 2√6.

⚠️ COMMON MISTAKE: Students often confuse the scaling factor with the transformation matrix, and incorrectly apply the matrix directly to the original dimensions of the rectangle, without considering the shape and area scaling.