16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.2.14

Determinants

Mathematics — Linear Algebra · cayley-hamilton-theorem-2x2

Problem

Let

A

be a

2 \times 2

matrix over a field

F

. Then the set of all matrices of the form

f(A)

, where

f

is a polynomial over

F

, is a commutative ring

K

with identity. If

B

is a

2 \times 2

matrix over

K

, the determinant of

B

is then a

2 \times 2

matrix over

F

, of the form

f(A)

. Suppose

I

is the

2 \times 2

identity matrix over

F

and that

B

is the

2 \times 2

matrix over

K

B = \begin{bmatrix} A - A_{11}I & -A_{12}I \\ -A_{21}I & A - A_{22}I \end{bmatrix}.

Show that

\det B = f(A)

, where

f = x^2 - (A_{11} + A_{22})x + \det A

, and also that

f(A) = 0

.

Related theory

The Cayley Hamilton Theorem

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