he eigen value of ⎡⎣⎢111111111⎤⎦⎥[111111111] are A. 0,0,0 B. 0,0,1 C. 0,0,3 D. 1,1,1

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

The given matrix is:

A = [1 1 1] [1 1 1] [1 1 1]

Subtract λI from A:

A - λI = [1-λ 1 1] [1 1-λ 1] [1 1 1-λ]

Now, we 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

Solving this cubic equation will give the eigenvalues of the matrix. The solutions to this equation are λ = 0, λ = 0, and λ = 3. Therefore, the eigenvalues of the given matrix are 0, 0, and 3. So, the correct answer is C. 0,0,3.