To find the eigenvalues of a symmetric matrix, we can follow these steps:

1. Write down the given matrix:
4 1 1
1 4 1
1 1 4

2. Set up the equation to find the eigenvalues:
|A - λI| = 0
where A is the given matrix, λ is the eigenvalue, and I is the identity matrix.

3. Subtract λ from the diagonal elements of the matrix:
(4-λ) 1 1
1 (4-λ) 1
1 1 (4-λ)

4. Expand the determinant of the resulting matrix:
(4-λ)((4-λ)(4-λ) - 1) - 1(1 - 1(4-λ)) + 1(1 - 1(4-λ))

5. Simplify the determinant expression:
(4-λ)(16 - 8λ + λ^2 - 1) - (4-λ) + (4-λ)
(4-λ)(15 - 8λ + λ^2) - 4 + λ

6. Expand and simplify further:
60 - 32λ + 4λ^2 - 15λ + 8λ^2 - λ^3 - 4 + λ
-λ^3 + 12λ^2 - 47λ + 56

7. Set the expression equal to zero and solve for λ:
-λ^3 + 12λ^2 - 47λ + 56 = 0

8. Use numerical methods or factorization techniques to find the eigenvalues. In this case, the eigenvalues are:
λ1 ≈ 1.536
λ2 ≈ 3.732
λ3 ≈ 6.732

Therefore, the eigenvalues of the given symmetric matrix are approximately 1.536, 3.732, and 6.732.

Question

To find the eigenvalues of a symmetric matrix, we can follow these steps:

1. Write down the given matrix:
   4  1  1
   1  4  1
   1  1  4

2. Set up the equation to find the eigenvalues:
   |A - λI| = 0
   where A is the given matrix, λ is the eigenvalue, and I is the identity matrix.

3. Subtract λ from the diagonal elements of the matrix:
   (4-λ)  1     1
   1     (4-λ)  1
   1      1    (4-λ)

4. Expand the determinant of the resulting matrix:
   (4-λ)((4-λ)(4-λ) - 1) - 1(1 - 1(4-λ)) + 1(1 - 1(4-λ))

5. Simplify the determinant expression:
   (4-λ)(16 - 8λ + λ^2 - 1) - (4-λ) + (4-λ)
   (4-λ)(15 - 8λ + λ^2) - 4 + λ

6. Expand and simplify further:
   60 - 32λ + 4λ^2 - 15λ + 8λ^2 - λ^3 - 4 + λ
   -λ^3 + 12λ^2 - 47λ + 56

7. Set the expression equal to zero and solve for λ:
   -λ^3 + 12λ^2 - 47λ + 56 = 0

8. Use numerical methods or factorization techniques to find the eigenvalues. In this case, the eigenvalues are:
   λ1 ≈ 1.536
   λ2 ≈ 3.732
   λ3 ≈ 6.732

Therefore, the eigenvalues of the given symmetric matrix are approximately 1.536, 3.732, and 6.732.

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