Quiz 9 :: Math 4A
Name (PRINT):
Last, First TARDIS
1) For parts (a)-(e) consider the following matrix
(a) (1 point) Compute the characteristic polynomial of A and use it to determine the
eigenvalues of A.
(b) (1 point) State the algebraic multiplicity of each eigenvalue. Before we determine
the eigenspaces ker(A I) for each eigenvalue , can we determine if the eigen-
vectors of A will form. an \eigenbasis" for R3? Why or why not?
(c) (1 point) For each eigenvalue , determine bases for the eigenspaces ker(A I).
(d) (1 point) State the geometric multiplicity of each eigenvalue. Do the eigenvectors
of A form. an eigenbasis for R3?
(e) (1 point) Can we determine if A is invertible solely by analyzing the eigenvalues or
eigenspaces of A? If so, determine if A is invertible. If not, explain why.
(f) (1 point) Try to form. a conjecture which relates the algebraic and geometric multi-
plicity of the eigenvalues of any n n matrix B, and the existence of an eigenbasis
for Rn corresponding to B.