Homework 2
YOUR NAME
Instructions: You can either print out the homework and write your answer on it(recommended),
or other ways that clearly shows your work and computations. Write your UNI in the right top
corner of each page. and your name in the left top corner. For computational excercise, circle the
final answer. This homework is due at 4:00pm June, 1st.
Problem 1. Compute in mind:



1
1 2 5

1 3  0 1 0  =
1
1 2 1
Problem 2. Compute the following expression:
1 2
1 1
(1)
3 1
2 1
(2)
1 2
3 1
0 3
1 4
+
2 5
1 4
1 1
3 0
Date: Printed on:May 28, 2015;
1
Homework 2
6 2
1 1
(3)
1 0
3 0

1
3
(4) So you know 

Your UNI:
2
1
2
1
6
1

5
4

2
0

1
2


1
1
0
1
1
3

3
4
=
1
0
Problem 3. Using basic row transformation and column transformation to compute the rank of
the following matrices, which of them are invertible?

1
3
(1) 
1
0
4
6
1
2
2
2
1
1

3
1

1
3


1 2 3
(2)  1 1 1 
8 9 10
2
YOUR NAME
Your UNI:
Problem 4. In this question, we consider the rank of block diagnal matrix.
Fix a partition P = (n1 , n2 ); n1 + n2 = n
Consider
the block square matrix
An1 ×n1
M=
Bn2 ×n2
Suppose r(An1 ×n1 ) = r1 ; r(Bn2 ×n2 ) = r2 , which means there exists invertible matrices Pn1 ×n1 ,
Qn1 ×n1 , such that
Ir1
P AQ =
0n1 −r1
.
And Rn2 ×n2 , Sn2 ×n2 , such that
Ir2
RBS =
0n2 −r2
(1) Using P,Q,R,S construct an invertible block matrix Un×n , Vn×n , such that


Ir1


0n1 −r1

UMV = 


Ir2
0n2 −r2
(2) Show that there rank of UMV
Doing the column and row transfor above is r1 + r2 (Hint:
Ir1 +r2
mation to the standard form
)
0n−r1 −r2
An1 ×n1
(3) With previous work, show that rank(
3
Bn2 ×n2
) = rank(An1 ×n1 )+rank(Bn2 ×n2 )
Homework 2
Your UNI:
Problem 5. Now we want to prove rank(AB) ≤ min{rank(A), rank(B)}
Ir
Ir
(1) Show that rank(
B) ≤ r and rank(A
) ≤ r[Hint: How many
0n−r
0n−r
non-zero rows/columns does it have?]
(2) prove rank(AB) ≤ min{rank(A),
rank(B)}
[Hint, suppose rank(A)=r, then there exists inIr
vertible P,Q such that P AQ =
, then rank(AB)=rank(PAB)=rank(P AQQ−1 B),
0n−r
explain why these equality is right and then use 1]
Problem 6. We already showed in the class that rank A B ≥min{rank(A),rank(B)}, Now
we like to use this theorem
+ B) ≤rank(A) + rank(B)
to showthat rank(A
A
A A+B
(1) Show that the rank of
and
are the same
B
B
A A+B
(2) Arrange the following number in descending order: rank
, rank A A + B ,
B
rank(A+B)
4
YOUR NAME
Your UNI:
(3) Write the complete proof of rank(A + B) ≤ rank(A) + rank(B)
5