Linear Algebra Done Right · Problem 9C.20

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

A

is an

n

-by-

n

matrix, and suppose

c

is such that

|A_{j,k}| \leq c

for all

j, k \in \{1, \ldots, n\}

. Prove that

|\det A| \leq c^n n^{n/2}.

The formula for the determinant of a matrix (9.46) shows that

|\det A| \leq c^n n!

. However, the estimate given by this exercise is much better. For example, if

c = 1

and

n = 100

, then

c^n n! \approx 10^{158}

, but the estimate given by this exercise is the much smaller number

10^{100}

. If

n

is an integer power of

2

, then the inequality above is sharp and cannot be improved.