Kunze Linear Algebra · Problem 5.2.3

Determinants

Mathematics — Methods · classical-adjoint

Problem

Let

K

be a commutative ring with identity. If

A

is a

2 \times 2

matrix over

K

, the classical adjoint of

A

is the

2 \times 2

matrix

\text{adj}\, A

defined by

\text{adj}\, A = \begin{bmatrix} A_{22} & -A_{12} \\ -A_{21} & A_{11} \end{bmatrix}.

\det

denotes the unique determinant function on

2 \times 2

matrices over

K

, show that ((a))

(\text{adj}\, A)A = A(\text{adj}\, A) = (\det A)I

; ((b))

\det(\text{adj}\, A) = \det(A)

; ((c))

\text{adj}(A^t) = (\text{adj}\, A)^t

. (

A^t

denotes the transpose of

A