Let 𝐴 𝑏𝑒 𝑎𝑛 𝑛 × 𝑛 matrix. The characteristic polynomial of A is a polynomial of degree

Let 𝜆 is an Eigen value of a 𝑛 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴.

The characteristic polynomial of a 3×33×3 matrix can be factorised as (2−λ)2(5−λ)(2−𝜆)2(5−𝜆). Which of the following statements are true? Select all that apply.The geometric multiplicity of the eigenvalue λ=𝜆= 55 could be 22The number of linearly independent eigenvectors for the eigenvalue λ=𝜆= 55 must be 11The geometric multiplicity of the matrix cannot be greater than 22The geometric multiplicity of the eigenvalue λ=𝜆= 22 could be 22The algebraic multiplicity of the matrix is 22The algebraic multiplicity of the eigenvalue λ=𝜆= 22 cannot be determined from the information we are givenThe eigenvalue λ=𝜆= 22 could have infinitely many eigenvector directionsThe algebraic multiplicity of the eigenvalue λ=𝜆= 22 is 2

) If A is nilpotent (Am = 0 for some m ≥ 1)then all eigenvalues of A are 0

The determinant rank of an 𝑚×𝑛 matrix A is equal to the ___ of A

If 0 is an Eigen value of a matrix A, then the set of columns of A is