Input interpretation
cobalt(III) ions | formal charges
Result
(3)
Table
3
Characteristic polynomial
3 - λ
Possible intermediate steps
Find the characteristic polynomial of the matrix M with respect to the variable λ: M = (3) To find the characteristic polynomial of a matrix, subtract a variable multiplied by the identity matrix and take the determinant: left bracketing bar M - λ I right bracketing bar left bracketing bar M - λ I right bracketing bar | = | left bracketing bar 3 - λ 1 right bracketing bar | = | left bracketing bar 3 - λ right bracketing bar invisible comma = left bracketing bar 3 - λ right bracketing bar The determinant of a diagonal matrix is the product of its diagonal elements: left bracketing bar 3 - λ right bracketing bar 3 - λ = 3 - λ: Answer: | | 3 - λ
Eigenvalues
λ_1 = 3
Possible intermediate steps
Find all the eigenvalues of the matrix M: M = (3) Find λ element C such that M v = λ v for some nonzero vector v: M v = λ v The only value of λ for which M v = λ v for any nonzero v is 3: Answer: | | 3
Eigenvectors
v_1 = (1)
Possible intermediate steps
Find all the eigenvalues and eigenvectors of the matrix M: M = (3) Find λ element C such that M v = λ v for some nonzero vector v: M v = λ v The only value of λ for which M v = λ v for any nonzero v is 3: 3 The equation M v = λ v is satisfied by each v element C^1, which means a suitable eigenvalue/eigenvector pair is: Answer: | | Eigenvalue | Eigenvector 3 | (1)