Worksheet on topic 3 - Matrices, determinants, and invertibility Math 165 - 10/22
1 0
1. (VTRMC 1, 2004) Let I denote the 2 × 2 identity matrix
and let
0 1
M=
I A
I B
,N =
,
B C
A C
where A, B, C are arbitrary 2 × 2 matrices with entries in R, the real numbers. Thus M
and N are 4 × 4 matrices with entries in R. Is it true that M is invertible (i.e. there is a
4 × 4 matrix X such that M X = XM = the identity ) implies that N is invertible? Justify
your answer.
2. (VTRMC 1, 2010) Let d be a positive integer and let A be a d × d matrix with integer
entries. Suppose that I + A + A2 + · · · + A100 = 0 (where I denotes the identity d × d matrix,
so I has 1’s on the main diagonal, and 0 denotes the zero matrix, which has all entries 0).
Determine the positive integers n ≥ 100 for which An + An+1 + · · · + A100 has determinant
±1.
3. (VTRMC 5, 2009) Let C denote the complex numbers and let M3 (C) denote the 3 by 3
matrices with entries in C. Suppose A, B ∈ M3 (C), B 6= 0 and AB = 0 (where 0 denotes
the 3 by 3 matrix with all entries zero). Prove that there exists 0 6= D ∈ M3 (C) so that
AD = DA = 0.
4. (VTRMC 6, 2007) Let n be a positive integer, let A, B be square symmetric n×n matrices
with real entries (if aij are the entries of A, the aij are real numbers, and aij = aji ). Suppose
there are n × n matrices X and Y (with complex entries) such that det(AX + BY ) 6= 0.
Prove that det(A2 + B 2 ) 6= 0 (det indicates the determinant).
5. (Putnam A2, 2008) Alan and Barbara play a game in which they take turns filing entries
of an initially empty 2008 × 2008 array. Alan plays first. At each turn, a player chooses a
real number and places it in a vacant entry. The game ends when all entries are filled. Alan
wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which
player has a winning strategy?
1
2
6. (Putnam A2, 2014) Let A be the n × n matrix whose entry in the i-th row and j-th
column is
1
min(i, j)
for 1 ≤ i, j ≤ n. Compute det(A).
7. (Putnam A3, 2009) Let dn be the determinant of the n × n matrix whose entries, from
left to right, and then top to bottom, are cos 1, cos 2, . . . , cos n2 . (For example,
cos 1 cos 2 cos 3
d3 = cos 4 cos 5 cos 6 .
cos 7 cos 8 cos 9
The argument of cos is always in radians, not degrees.) Evaluate limn→∞ dn .
8. (Putnam A3, 2004) Define a sequence by {un }∞
n=0 by u0 = u1 = u2 = 1, and thereafter
by the condition taht
un un+1
det
= n!
un+2 un+3
for all n ≥ 0. Show that un is an integer for all n. (By convention, 0! = 1.)
9. (Putnam A4, 2002) In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty 3 × 3
matrix. Palyer 0 counters with a 0 in a vacant position, and play continues until the 3 × 3
matrix is completed with five 1’s and four 0’s. Player 0 wins if the determinant is zero, and
Player 1 wins otherwise. Assume both players pursue optimal strategies, who will win and
how?
10. (Putnam B6, 2005) Let Sn denote the set of all permutations of the numbers 1, 2, . . ., n.
For π ∈ Sn , let σ(π) = 1 if π is even permutation and σ(π) = −1 if π is an odd permutation.
Also, let ν(π) denote the number of fixed points of π. Show that
X σ(π)
n
= (−1)n+1
.
ν(π)
+
1
n
+
1
π∈S
n