Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest. Do not list the same eigenvalue multiple times.)4 1 1 1 4 11 1 4𝜆i =

To find the eigenvalues of a symmetric matrix, we need to solve the characteristic equation. The characteristic equation is given by det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix.

In this case, the given matrix is:

4 1 1 1 4 1 1 1 4

To find the eigenvalues, we subtract λI from the matrix A:

(4-λ) 1 1 1 (4-λ) 1 1 1 (4-λ)

Next, we calculate the determinant of this matrix:

det(4-λ 1 1 1 4-λ 1 1 1 4-λ) = 0

Expanding the determinant, we get:

(4-λ)((4-λ)(4-λ) - 1) - (1)((1)(4-λ) - (1)(1)) + (1)((1)(1) - (4-λ)(1)) = 0

Simplifying further, we have:

(4-λ)((16-8λ+λ^2) - 1) - (1)(3-λ) + (1)(1-(4-λ)) = 0

(4-λ)(15-8λ+λ^2) - (3-λ) + (1-4+λ) = 0

Expanding and simplifying, we get:

60 - 32λ + 4λ^2 - 15 + 8λ - λ^2 - 3 + λ + 1 - 4 + λ = 0

Combining like terms, we have:

4λ^2 - 24λ + 43 = 0

To solve this quadratic equation, we can use the quadratic formula:

λ = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 4, b = -24, and c = 43. Plugging these values into the quadratic formula, we get:

λ = (-(-24) ± √((-24)^2 - 4(4)(43))) / (2(4))

Simplifying further, we have:

λ = (24 ± √(576 - 688)) / 8

λ = (24 ± √(-112)) / 8

Since the discriminant is negative, the eigenvalues are complex numbers. Therefore, the eigenvalues of the given symmetric matrix are:

λ1 = (24 + √(-112)) / 8 λ2 = (24 - √(-112)) / 8

However, since the question asks for the eigenvalues from smallest to largest and does not list the same eigenvalue multiple times, we can conclude that the eigenvalues of the given symmetric matrix are:

λ1 = (24 + √(-112)) / 8 λ2 = (24 - √(-112)) / 8