Write down detailed proofs of every statement you make 1. A 4 × 4 matrix A has eigenvalues λ1 = 1,λ2 = 0,λ3 = 2,λ4 = −1.
• Is A invertible? Why or why not?
• Is A diagonalizable? Why or why not?
• Find the the characteristic polynomial, the trace and determinant of A.
2. Find a relation between the eigenvalues of a non-singular matrix A and those of its inverse A−1
3. Find a relation between the eigenvalues of a matrix A and those of its square A2 = AA
4. For the following either find an example or prove that such examplecannot exist:
(a) A 4×4 matrix with eigenvalues λ1 = 1 with algebraic multiplicity 2 and geometric multiplicity 1; λ2 = 2 with algebraic multiplicity 1 and geometric multiplicity 1 and λ3 = 3 with algebraic multiplicity 1 and geometric multiplicity 1.
(b) A 4×4 matrix with eigenvalues λ1 = 1 with algebraic multiplicity 1 and geometric multiplicity 2; λ2 = 2 with algebraic multiplicity 2 and geometric multiplicity 1 and λ3 = 3 with algebraic multiplicity 1 and geometric multiplicity 1.
(c) A 4×4 matrix with eigenvalues λ1 = 1 with algebraic multiplicity 2 and geometric multiplicity 1; λ2 = 2 with algebraic multiplicity 2 and geometric multiplicity 1 and λ3 = 3 with algebraic multiplicity 1 and geometric multiplicity 1.
(d) A 4 × 4 matrix with one eigenvalue λ1 = π with algebraic multiplicity 4 and geometric multiplicity 1;
5. Construct a 3×3 matrix A with eigenvalues π,π2,π3 and corresponding eigenvectors (1,0,1), (1,1,0), (0,0,1). Is such matrix unique?
