Final answer:
To find the area of shape Q after a linear transformation, we first apply the transformation to the vectors defining parallelogram P, and then calculate the determinant of the matrix formed by the transformed vectors. The absolute value of the determinant gives us the area of the new parallelogram.
Step-by-step explanation:
The question asks us to find the area of shape Q which is obtained by applying the linear transformation T(x) = [3 1; 1 2]x to the parallelogram P determined by the vectors [4; 1] and [3; -1]. To solve this, we apply the transformation T to each vector and then use the resultant vectors to find the area of the new parallelogram Q.
First, we apply the transformation T to the vectors that define parallelogram P:
- For vector [4; 1]: T([4; 1]) = [3 1; 1 2] * [4; 1] = [12+1; 4+2] = [13; 6]
- For vector [3; -1]: T([3; -1]) = [3 1; 1 2] * [3; -1] = [9-1; 3-2] = [8; 1]
The area of the new parallelogram Q can be found by the determinant of the matrix formed by the transformed vectors:
Area of Q = |13 8|
| 6 1| = (13*1 - 8*6) = 13 - 48 = -35
The area is the absolute value of the determinant, so the area of Q is 35 square units.