Course 111: Algebra, 20th April 2007
To be handed in at tutorials on April 23rd and 24th.
1. Consider the matrix


1 0
1

A =  2 −2 −1 

0 0
1
Write A in its Jordan form, J by determining the matrix V which
transforms A according to J = V −1 AV .
Use the formula in your notes to determine e2J .
2. Consider the matrix equation, Ax = b with
A=
1
1
1 1+
!
, 0 < << 1
(1)
Determine the condition number of the matrix A with respect to the
∞ norm.
Is the matrix well or poorly conditioned? Given
2
2
b=
!
,
determine a solution, x to the matrix equation Ax = b.
Determine what happens to the solution when
A=
1 1
1 1
!
, 0 < << 1
and b is defined as above.
Also, determine the effect of a small change in b on the solution by
considering solutions when b = (2, 2)T and b = (2 + , 2)T and A is as
given in Eqn 1.