16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.2.11

Determinants

Mathematics — Linear Algebra · determinant-product-rule

Problem

Let

K

be a commutative ring with identity, and let

D

be an alternating 2-linear function on

2 \times 2

matrices over

K

. Show that

D(A) = (\det A)D(I)

for all

A

. Now use this result (no computations with the entries allowed) to show that

\det(AB) = (\det A)(\det B)

for any

2 \times 2

matrices

A

and

B

over

K