Final answer:
The characteristic polynomial of the matrix A is -λ^3 + 12λ^2 - 47λ + 60. The eigenvalues are the solutions to setting this polynomial equal to zero.
Step-by-step explanation:
To find the characteristic polynomial of a matrix, we must first subtract λ (the eigenvalue) times the identity matrix from our original matrix and then take the determinant of the resulting matrix. For the matrix A = [5 4 2, 4 5 2, 2 2 2], the process looks like this:
A - λI =
\[
\begin{bmatrix}
5-λ & 4 & 2 \\
4 & 5-λ & 2 \\
2 & 2 & 2-λ
\end{bmatrix}
\]
To find the determinant, we'll expand out to get the characteristic polynomial:
P(λ) =
\[
(5-λ)((5-λ)(2-λ) - 4) - 4(4(2-λ) - 4) + 2(8 - 2(5-λ))
\]
Expanding and simplifying, we get the characteristic polynomial in expanded form:
P(λ) = -λ^3 + 12λ^2 - 47λ + 60
To find the eigenvalues, we set P(λ) equal to zero:
-λ^3 + 12λ^2 - 47λ + 60 = 0
The eigenvalues are the roots of this polynomial, which can often be found using methods such as factoring, completing the square, or applying the quadratic formula. For higher-degree polynomials, numerical methods or software tools may be used.