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 = 3, 16 For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.)dim(xi) =
Question
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 = 3, 16 For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.)dim(xi) =
Solution
To find the eigenvalues of the symmetric matrix, we can start by finding the determinant of the matrix subtracted by 𝜆 times the identity matrix.
The given matrix is: 4 1 1 1 4 1 1 1 4
Subtracting 𝜆 times the identity matrix: (4-𝜆) 1 1 1 (4-𝜆) 1 1 1 (4-𝜆)
Next, we can find the determinant of this matrix and set it equal to zero to find the eigenvalues.
Determinant = (4-𝜆)((4-𝜆)(4-𝜆) - 1) - 1(1 - 1(4-𝜆)) + 1(1 - 1(4-𝜆)) Determinant = (4-𝜆)((4-𝜆)^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)((16 - 8𝜆 + 𝜆^2) - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(15 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(15 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(14 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(14 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(13 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(13 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(12 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(12 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(11 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(11 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(10 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(10 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(9 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(9 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(8 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(8 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(7 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(7 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(6 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(6 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(5 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(5 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(4 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(4 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(3 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(3 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(2 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(2 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(1 - 8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(1 - 8𝜆 + 𝜆^2 - 1) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - (4-𝜆) + (4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-𝜆)(-8𝜆 + 𝜆^2) - 2(4-𝜆) + 2(4-𝜆) Determinant = (4-
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