16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.4.9

Determinants

Mathematics — Linear Algebra · determinant-rank

Problem

Let

A

be an

n \times n

matrix over a field,

A \neq 0

. If

r

is any positive integer between

1

and

n

, an

r \times r

submatrix of

A

is any

r \times r

matrix obtained by deleting

(n-r)

rows and

(n-r)

columns of

A

. The determinant rank of

A

is the largest positive integer

r

such that some

r \times r

submatrix of

A

has a non-zero determinant. Prove that the determinant rank of

A

is equal to the row rank of

A

(

=

column rank

A