1) Find the characteristic polynomial and the eigenvalues of the following
matrices over C.
3 2 2
1 4 1
2) Let A € Mn(C) and a € C. Using the definition of the determinant given
in class, prove that if B is the matrix obtained by multiplying a single row
of A by a. then
det(B) = det(A).
3) (#14, Section 5.1) For any square matrix A. prove that A and At have
the same characteristic polynomial (and hence the same eigenvalues). You
may assume that A € Mn(C) or Mn(R).
4) a) If A and B are two similar n X 72 matrices, prove that A and B have
the same characteristic polynomial, and hence, the same eigenvalues.
b) Show that if T: Cr C" is a linear map, then the eigenvalues of [2%
and the eigenvalues of  are the same for any two ordered bases 8.7 of C".
Conclude that the eigenvalues of a linear transformation are independent of
the matrix form given by a choice of basis.
5) Let A € Mn(C) and suppose A is invertible. Note that zero is not an
eigenvalue of A.
a) Prove that l is an eigenvalue of A if and only if 1/X is an eigenvalue
b) Prove that A is diagonalizable if and only if A-1 is diagonalizable.
6) Let A =
€ M2(R). Define a map HA M2(R) - M2(R) by
for all B
a) Prove that HA is a linear map.
b) Let A =
Find the eigenvalues of HA.
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