Math 223, Section 101—Homework #7
due Friday, November 4, 2011 at the beginning of class
Practice problems. The usual rules apply.
1. I have linked to a web page with practice problems about complex numbers. You can go
to the URL http://tinyurl.com/MATH223 or follow the link from the course web
page. I’m sure there are lots of similar practice problems out there.
2. Friedberg, Insel, and Spence, Section 4.2, pp. 220–1, #1 (change every F to R) and #3
3. Friedberg, Insel, and Spence, Section 4.3, p. 228, #1(a)–(f) (change Mn×n (F ) to Mn×n (R))
4. Friedberg, Insel, and Spence, Section 4.3, p. 229, #10 and #12
5. Friedberg, Insel, and Spence, Section 4.3, p. 231, #26(a),(c),(e),(g)
6. Friedberg, Insel, and Spence, Section 4.4, p. 236, #1 (change Mn×n (F ) to Mn×n (R)) and
#2(a),(c)
7. Friedberg, Insel, and Spence, Section 4.4, pp. 236–7, #3(a),(c),(e),(g) and 4(a),(c),(e),(g)
8. If two matrices A and B are similar, prove that det(A) = det(B).
Homework problems. The usual rules apply.
1. Friedberg, Insel, and Spence, Section 4.4, pp. 236–7, #3(b),(d),(f),(h)
2. Friedberg, Insel, and Spence, Section 4.4, pp. 236–7, #4(f),(h). For (f) you can use whatever method you want; for (h), calculate the determinant by using the forward pass of
Gaussian elimination to transform the matrix into row echelon form.
3. (a) Solve the following system of linear equations over C:
z1 + (−2)z2 +
(−5i)z3 = −2
−2z1 + (3 + 3i)z2 + (−7 + 11i)z3 = 2i
−iz1 +
z2 +
0 =3+i
That is, find all ordered triples (z1 , z2 , z3 ) of complex numbers such that the above
equations are satisfied. (It’s easy to make mistakes in the calculations: check your
answer!) 

1
−2
−5i
(b) What is det  −2 3 + 3i −7 + 11i ? Why? (This should be a one-liner.)
−i
1
0
4. Friedberg, Insel, and Spence, Section 4.3, p. 230, #24. The answer is in the back of the
book, but you will need to justify why it is the right answer.
5. Call an n × n matrix “backwards-upper-triangular” if all of its entries below the northeastto-southwest diagonal equal 0; call it “backwards-lower-triangular” if all of its entries
above the northeast-to-southwest diagonal equal 0. For example, when n = 3, these matrices look like




0 0 ∗
∗ ∗ ∗
 ∗ ∗ 0  and  0 ∗ ∗ ,
∗ 0 0
∗ ∗ ∗
respectively, where each ∗ denotes any real number. Suppose that A is a backwards-uppertriangular matrix such that all of the entries on the northeast-to-southwest diagonal are
nonzero. Prove that A is invertible and that A−1 is a backwards-lower-triangular matrix.
6. Let A and B be invertible n × n matrices. Let α be the subset of Rn consisting of the n
columns of A (ordered from left to right), and let β be the subset of Rn consisting of the
n columns of B (ordered from left to right). Prove that α and β are ordered bases for Rn ,
and prove that the change of coordinates matrix from β to α is A−1 B.
The next two problems deal with “block matrices”,
which
matricesthat aremade bystacking
are 1 2
5 6
9 9
other matrices together. For example, if A =
,B =
,C =
, and O2
3 4
7 8
9 9
is the 2 × 2 zero matrix, then


1
2
5
6
 3 4 7 8 
A B

=
 9 9 0 0 .
C O2
9 9 0 0
(In general, the submatrices in block matrices could be different sizes; however, in this homework,
we’ll only deal with the case where they’re all square matrices of the same size.) Note: you can
use the identity (∗) from the Bonus question below, even if you don’t prove it!
7. In this problem, A, B, and C are all n × n matrices (with the same n), while In denotes the
n × n identity matrixand On denotes
the n × n zero matrix.
A On
(a) Prove that rank
= rank(A) + rank(B). (Hint: number of linearly
On B
independent rows
or columns.)
On A
(b) Prove that rank
≥ rank(A) + rank(B).
B C
In A
In On
(c) Prove that
and
are always invertible. Hint: use (∗) to guess
On In
A In
their inverses.
8. Let A, B, C, In , On be as in the previous problem. (You can use results from the previous
problem in yoursolution tothis
one.)
In A
−ABC On
In On
On AB
(a) Prove that
=
.
On In
On
B
C In
BC B
(b) Prove that rank(AB) + rank(BC) ≤ rank(ABC) + rank(B).
(c) Let V be a finite-dimensional vector space, and let S, T, U : V → V be three linear
transformations from V to V . Prove that rank(U T ) + rank(T S) ≤ rank(U T S) +
rank(T ).
Bonus question. Suppose that A, B, C, D, E, F, G, H are square matrices all of the same size.
Prove that
A B
E F
AE + BG AF + BH
=
.
(∗)
C D
G H
CE + DG CF + DH