Final answer:
To find the area of the transformed ellipse Aᵗ, we need to find the eigenvalues and eigenvectors of matrix A. The area of Aᵗ is given by the formula A = π * (λ₁ * λ₂). Substituting the values of λ₁ and λ₂, we can calculate the area of Aᵗ.
Step-by-step explanation:
To find the area of the transformed ellipse Aᵗ, we need to find the eigenvalues and eigenvectors of matrix A. The eigenvalues are the solutions to the characteristic equation det(A - λI) = 0, where I is the identity matrix. The eigenvectors corresponding to the eigenvalues λ₁ and λ₂ are v₁ and v₂.
The area of the transformed ellipse Aᵗ is given by the formula A = π * (λ₁ * λ₂).
Substituting the values of λ₁ and λ₂, we can calculate the area of Aᵗ.
The question is asking to compute the area of an ellipse Aᵗ after a linear transformation represented by the matrix A = [[2, 1], [1, 3]]. To find the area of the transformed ellipse, we need to use the determinant of the matrix A, which scales the area of the ellipse. Since the area of the original ellipse is given as 5, we calculate the determinant of matrix A which is (2*3 - 1*1) = 5. The area of the transformed ellipse Aᵗ will be the original area times the determinant of matrix A, giving us 5 * 5 = 25.