Question

Let be a 3 × 3 matrix with real entries such that det() = 6 and the trace of

is 0. If det( + ) = 0, where denotes the 3 × 3 identity matrix, then find

the eigen values of .​

Answer 1

Answer:

Given that,

N be a 3 by 3 matrix with real number.

Let the diagram values of N be λ1, λ2 and λ3

Properties of Eigen values:

(i) If λ is an Eigen value of a matrix ‘A’ then λ2 will be an Eigen value of the matrix A2.

(ii) Product of Eigen values is equal to determinant.

(iii) Sum of the Eigen values is equal to trace.

As N2 = 0

⇒ |N2| = 0

Determinant of N2 = 0

Eigen values of N2 are

λ12,λ22andλ32

⇒λ12λ22λ32=0⇒λ1λ2λ3=0

And, λ1 + λ2 + λ3 = trace

Since, N2 = 0, N is nilpotent matrix.

For a nilpotent matrix, determinant and trace is zero.

⇒ λ1 + λ2 + λ3 = 0 ----(1)

And

λ12.λ22.λ32=0 ----(2)

From the above two equations,

λ1 = 0 and λ2 = -λ3

From the options, λ1 = 0, λ2 = 0, λ3 = 0

Eigen values of N are = 0, 0, 0