[SOLVED] MA502 Homework 6

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 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.
   • 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.
   • 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.
   • 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?