MTH5112 Linear Algebra I 2012-2013
Coursework 2
Please hand in your solutions of the starred feedback exercises by noon on Friday 12 October
using the red Linear Algebra I Collection Box in the Basement. Don’t forget to put your name (with
your surname underlined) and student number on your solutions, and to staple them. Please also
indicate the day and time of the tutorial at which you will collect your marked work.
Exercise 1. Give a geometric explanation of why a homogeneous linear system consisting of two
equations in three unknowns must have infinitely many solutions.
Exercise 2. Consider the matrices Aand Bgiven by
A=2 4
−6 0, B =1−5
−3 2 .
Find (a) A−B, (b) 1
2A+ 3B.
Exercise* 3. For the following matrices decide which products of two of them exist (including the
possibility that a matrix is multiplied by itself) and calculate these products.
A=
−2 0
7−1
0 3
, B =
−110
2 0 3
0−2 1
.
Exercise 4. Let Aand Bbe square matrices of the same size. Show that
(A+B)2=A2+ 2AB +B2
if and only if Aand Bcommute.
Exercise 5. Show that the sum and the product of two upper triangular matrices of the same size
is again upper triangular.
Exercise 6. Let Aand Bbe matrices of the same size. Show the following:
(a) (αA)T=αAT;
(b) (A+B)T=AT+BT.
Exercise* 7. Let Aand Bbe square matrices of the same size. Show the following:
(a) If A3=Othen (I+A)is invertible and (I+A)−1=I−A+A2.
(b) If Aand Bare symmetric then AB is symmetric if and only if Aand Bcommute.
Exercise 8. Show that if Ais invertible then ATis invertible and
(AT)−1= (A−1)T.
A cash prize is offered for the best solution received to the following question. Do not hand it in
with your coursework, but hand it to me in person at the Wednesday lecture (17th Oct) or before.
Prize question. A matrix Pis called an idempotent if P2=P. Give examples of idempotents.
Can an idempotent be invertible? If Pis an idempotent, is I−Pinvertible? What about I−aP ,
where ais a scalar?