MATH 395 PROBLEMS 2

IGOR KRIZ

Regular problems:

1. Calculate:

det ( \begin{matrix}1 & 0 & 3 & 1 \\ 2 & 2 & 6 & 0 \\ −1 & 0 & −3 & 1 \\ 4 & 1 & 12 & 0 \end{matrix} ) .

2. Find an explicit formula for

(a) The inverse of a general 3 × 3 matrix

(b) The solution of a general system of two linear equations with two unknowns. [Use determinants and Cramer’s rule.]

3. Find the values of x for which det(A) = 0, where

A = ( \begin{matrix}1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix} ) .

4. Find the sign of the permutation

(a) [2, 3, …, n]

(b) [n, n − 1, n − 2, ..., 1].

Challenge problem:

5. Let p(x) = x^{m} + a_{1}x^{m−1} + ... + a_{m}x^{0} and q(x) = x^{n} + b_{1}x^{n−1} + ... + b_{n}x^{0}. The resultant R(p, q) of the polynomials p(x), q(x) is the determinant of the (m + n) × (m + n) matrix

( \begin{matrix}1 & a_{1} & … & a_{m} \\ & 1 & a_{1} & … & a_{m} \\ & & … \\ & & 1 & a_{1} & … & a_{m} \\ 1 & b_{1} & … & b_{n} \\ & 1 & b_{1} & … & b_{n} \\ & & … \\ & & 1 & b_{1} & … & b_{n} \end{matrix} ) .

Let ζ_{1}, ..., ζ_{m} be the roots of the polynomial p(x), let ξ_{1}, ..., ξ_{n} be the roots of the polynomial q(x). Then we have

R(p, q) = ∏_{i<j}(ξ_{i} − ζ_{j}) (1)

(a) Prove formula (1) in the case n = 1. [Quite easy from definition.]

(b) Prove formula (1) for all m, n. [Much harder.]

(c) The discriminant of a polynomial p(x) is the resultant of p(x) and its derivative: R(p, p′). Write down the discriminant of polynomials of degree 2, 3 explicitly.

(d) Prove that the discriminant of a polynomial p(x) is non-zero if and only if p(x) has no root of multiplicity > 1.

(e) Prove that the discriminant of a polynomial is the square of the Vandermonde determinant of its roots. [What can you say about two polynomials with the same roots and multiplicities?]