16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.2.3

Determinants

Mathematics — Linear Algebra · classical-adjoint-2x2

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

\mathrm{adj}\, A

defined by

\mathrm{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))

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

; ((b))

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

; ((c))

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

. (

A^t

denotes the transpose of

A