Find the Spectrum of A. Find the algebraic and geometric multiplicity for only any repeated eigenvalue of the matrix A. [1000 0 1 5 -10 102 0 1 0 0 3

Find the eigenvalues and corresponding eigenvectors for the matrix 8 -8 -2 A 4 -3 -2 3-4 1 Write down the modal matrix M and spectral matrix A of A, and confirm that M-¹AM = A

Find bases for the eigenspaces associated with each eigenvalue of the matrix 3 0 0 -2 3-2 2 0 5 Find, if possible, a nonsingular matrix P such that P-¹AP is diagonal. A =

Let A be an nxn matrix. Determine whether the statement below is true or false. Justify the answer. The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A. The statement is false. The multiplicity of a root r of the characteristic equation of A is the number of eigenvectors corresponding to that root. The statement is true. It is the definition of the multiplicity of a root of the characteristic equation of A. The statement is true. It is the definition of the algebraic multiplicity of an eigenvalue of A. OThe statement is false. The multiplicity of a rootr of the characteristic equation. of A is called the geometric multiplicity of r as an eigenvalue of A.