Let the matrix M =4 0 00 4 00 1 5.(a) (3pts) Find the characteristic equation and the eigenvalues.(b) (2pts) Determine the multiplicity for each eigenvalue of the matrix M
Question
Let the matrix M =4 0 00 4 00 1 5.(a) (3pts) Find the characteristic equation and the eigenvalues.(b) (2pts) Determine the multiplicity for each eigenvalue of the matrix M
Solution
(a) The characteristic equation of a matrix is found by subtracting lambda (λ) from the diagonal entries and then taking the determinant. For matrix M, the characteristic equation is:
det(M - λI) = 0, where I is the identity matrix and λ represents the eigenvalues.
So, we have:
det(4-λ 0 0 0 4-λ 0 0 1 5-λ) = 0
This simplifies to (4-λ)^2 * (5-λ) = 0
Setting this equation to zero gives the roots λ1 = 4, λ2 = 4, λ3 = 5. These are the eigenvalues of matrix M.
(b) The multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic equation.
From the characteristic equation, we can see that the eigenvalue 4 appears twice and the eigenvalue 5 appears once. Therefore, the multiplicity of the eigenvalue 4 is 2 and the multiplicity of the eigenvalue 5 is 1.
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