The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities. 2) λ5 - 24λ4 + 189λ3 - 486λ2

The eigenvalues of the 5 x 5 matrix are 0 (multiplicity: 2), 6 (multiplicity: 1), 9 (multiplicity: 1) y 1 (multiplicity: 1).

The eigenvalues of the 5 x 5 matrix are represented by the roots of the characteristic polynomial. The multiplicity of a eigenvalue is the number of times that a root is repeated.

Now we proceed to determine the roots of the characteristic polynomial by algebraic means:

The eigenvalues of the 5 x 5 matrix are 0 (multiplicity: 2), 6 (multiplicity: 1), 9 (multiplicity: 1) y 1 (multiplicity: 1).

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The answer is below

Step-by-step explanation:

Since the highest degree of the characteristic polynomial is 5, this is a 5 × 5 matrix. The characteristic polynomial is given by:

P(λ) = λ⁵ - 24λ⁴ + 189λ³ - 486λ²

To find the eigen values and the multiplicities, we have to equate the characteristic polynomial to zero. i.e P(λ) = 0. Therefore:

λ⁵ - 24λ⁴ + 189λ³ - 486λ² = 0

λ²(λ³ - 24λ² + 189λ - 486) = 0

λ²(λ - 9)²(λ - 6) = 0

Therefore the eigen values are:

λ = 0 with multiplicity of 2

λ = 9 with multiplicity of 2

λ = 6 with multiplicity of 1

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step-by-step explanation: