Question

If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ³ – 4λ² + aλ + 30, a ∈ R, and one eigenvalue of M is 2, then the largest among the absolute value of the eigenvalues of M is __________.  

Answer 1

Concept:

Characteristic equation: If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix. The determinant of this matrix equated to zero i.e. |A – λI| = 0 is called the characteristic equation of A.

Eigenvalues: The roots of the characteristic equation are called Eigenvalues or latent roots or characteristic roots of matrix A.

Calculation:

Here, polynomial is: λ³ – 4λ² + aλ + 30, a ∈ R

One eigen value is 2. So, it must satisfy this equation.

⇒ 2³ – 4 × 2² + a × 2 + 30 = 0,

⇒ 8 – 16 + 2a + 30 = 0

⇒ - 8 + 2a = -30

⇒ a = -11

Now, the polynomial will be 

λ³ – 4λ² + (-11)λ + 30 = 0

λ³ – 4λ² – 11λ + 30 = 0

Factorising we get,

(λ - 2) (λ - 5) (λ + 3) = 0

⇒ λ = 2, 5, -3

The largest among the absolute value of the eigenvalues of M is 5.