Write down detailed proofs of every statement you make
1. Let A be a n×n matrix and let J be the set of polynomials f(t) ∈K(t) such that f(A) = 0. Prove that J is an ideal. Can you point out a specific polynomial of degree n and one of degree n2 in J?
2. For any n × n matrix define the cofactor matrix coA to be the n × n matrix whose (i,j) entry is (−1)i+j times the determinant of the (n − 1) × (n − 1) matrix obtained from A deleting the i−th row and j−th columns. Let the classical adjoint matrix ad(A) (also called adjugate or adjunct) be defined as the transpose of the cofactor matrix. Prove that Aad(A) = ad(A)A = det(A)I.
3. Let A be an upper triangular n × n matrix.
• Prove that all powers Ak are upper triangular.
• Derive a formula for the eigenvalues of f(A) when f ∈K(t) is a
polynomial.
• Find a relation between the eigenvalues of a non-singular matrix A and those of its inverse A−1
• Using the property above, find the eigenvalues and the characteristic polynomial of
(A3 − 3A2 + I)−1
where A is an upper triangular 3 × 3 matrix with eigenvalues 1,0,−1. (As part of the problem you will need to check that A3 − 3A2 + I is indeed invertible even if A is clearly not so)
4. If A is a square matrix with eigenvalues 1,2,3 find the eigenvalues of A100. Provide a detailed proof of your answer (note we are not assuming that A is 3 × 3).
