Assignment 4
1) Find the characteristic polynomial for the matrix
10 9 0 0
4 2 0 0
:0 0 2 7
0 0 1 2
A
−
−
=
− −
2) Prove that the characteristic polynomial of an n n× matrix A is always of degree
n.
3) Prove that if
{ }
1 2
, ,..., k
v v v
is a collection of eigenvectors associated to distinct
eigenvalues, then this collection is linearly independent.
4) Find the characteristic polynomial, eigenvalues, and eigenvectors of the following
matrices:
a)
1 2 2
: 1 2 1
1 1 0
A
− − −
=
− −
b)
2 2 3
: 2 3 2
4 2 5
A
−
= −
−
5)
a) Suppose that A is similar to an upper triangular matrix U. Prove that the
determinant of A is equal to the product of its eigenvalues (including
multiplicity). (Hint: start by finding out the relationship between det A
and det U.)
b) Geometrically, give an idea of what the determinant of a matrix measures
using the result in a).
6) Determine whether A is diagonalizable. If so, find the matrix P which
diagonalizes it and the diagonal matrix that it is similar to.
a)
0 0 2
: 3 4 0
3 1 3
A
−
= −
−
b)
1 4 2
: 3 4 0
3 1 3
A
− −
= −
−
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