The eigen value of ⎡⎣⎢111111111⎤⎦⎥[111111111]  are

The given matrix is a 3x3 matrix with all elements as 1. Let's denote this matrix as A.

A = ⎡⎣⎢111111111⎤⎦⎥

The eigenvalues of a matrix A are the roots of the characteristic equation, which is given by det(A - λI) = 0, where I is the identity matrix and λ represents the eigenvalues.

For matrix A, (A - λI) would be:

A - λI = ⎡⎣⎢1-λ1111111-λ1111-λ⎤⎦⎥

Now, we need to find the determinant of (A - λI) and set it equal to zero.

det(A - λI) = (1-λ)[(1-λ)(1-λ) - (1)(1)] - 1[(1)(1) - (1)(1)] + 1[(1)(1) - (1)(1)] = (1-λ)[(1-λ)^2 - 1] = (1-λ)[1 - 2λ + λ^2 - 1] = (1-λ)(λ^2 - 2λ) = λ^3 - 3λ^2 + 2λ

Setting this equal to zero gives the characteristic equation:

λ^3 - 3λ^2 + 2λ = 0

The roots of this equation are the eigenvalues of matrix A. This equation can be factored to:

λ(λ-1)(λ-2) = 0

Setting each factor equal to zero gives the solutions λ=0, λ=1, λ=2. Therefore, the eigenvalues of the given matrix are 0, 1, and 2.