MathDB

Do either

(1)

(2)

(1)

Let

A

be matrix

(a_{ij}), 1 \leq i,j \leq 4.

Let

d =

det

(A),

and let

A_{ij}

be the cofactor of

a_{ij}

, that is, the determinant of the

3 \times 3

matrix formed from

A

by deleting

a_{ij}

and other elements in the same row and column. Let

B

be the

4 \times 4

matrix

(A_{ij})

and let

D

be det

B.

Prove

D = d^3

(2)

Let

P(x)

be the quadratic

Ax^2 + Bx + C.

Suppose that

P(x) = x

has unequal real roots. Show that the roots are also roots of

P(P(x)) = x.

Find a quadratic equation for the other two roots of this equation. Hence solve

(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.

B1