[Solved] MA502 Homework 6

Write down detailed proofs of every statement you make 1. A 4 4 matrix A has eigenvalues ₁ = 1,₂ = 0,₃ = 2,₄ = 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 A¹
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 44 matrix with eigenvalues ₁ = 1 with algebraic multiplicity 2 and geometric multiplicity 1; ₂ = 2 with algebraic multiplicity 1 and geometric multiplicity 1 and ₃ = 3 with algebraic multiplicity 1 and geometric multiplicity 1.
   • A 44 matrix with eigenvalues ₁ = 1 with algebraic multiplicity 1 and geometric multiplicity 2; ₂ = 2 with algebraic multiplicity 2 and geometric multiplicity 1 and ₃ = 3 with algebraic multiplicity 1 and geometric multiplicity 1.
   • A 44 matrix with eigenvalues ₁ = 1 with algebraic multiplicity 2 and geometric multiplicity 1; ₂ = 2 with algebraic multiplicity 2 and geometric multiplicity 1 and ₃ = 3 with algebraic multiplicity 1 and geometric multiplicity 1.
   • A 4 4 matrix with one eigenvalue ₁ = with algebraic multiplicity 4 and geometric multiplicity 1;
5. Construct a 33 matrix A with eigenvalues ,²,³ and corresponding eigenvectors (1,0,1), (1,1,0), (0,0,1). Is such matrix unique?