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MA502 – Homework 7

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MA502 – Homework 7.
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 n
2
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
(A
3 − 3A
2 + 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).
1

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