Algebra II, MATH 251 Instructor: Dr. E. Goren.

Assignment 5

To be submitted by February 14, 12:00

1. Deduce from the theorems on determinants the following:

(1) If a column is zero, the determinant is zero.

(2) det(A) = det(A^{t}), where A^{t} is the transposed matrix.

(3) If a row is zero, the determinant is zero.

(4) Let A be a matrix in “upper diagonal block form”:

A = ( \begin{matrix}A_{1} & & & & & ⋆ \\ 0 & A_{2} \\ & & . & . & . \\ 0 & & & 0 & & A_{k} \end{matrix} ) .

Here each A_{i} is a square matrix say of size r_{i}, and A_{2} starts at the r_{1} + 1 column and r_{1} + 1 row, etc. Prove that

det(A) = det(A_{1}) det(A_{2}) ⋯ det(A_{k}).

Conclude that the determinant of an upper triangular matrix is given by

det ( \begin{matrix}a_{11} & & & & & ⋆ \\ 0 & a_{22} \\ & & . & . & . \\ 0 & & & 0 & & a_{kk} \end{matrix} ) = a_{11}a_{22} ⋯ a_{kk}.

(Here each a_{ii} is a scalar).

2. Calculate the following series of determinants.

(1) det ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , det ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) , det ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) , ⋯

(2) det ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) , det ( \begin{matrix}1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{matrix} ) , det ( \begin{matrix}1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ) , det ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix} ) , ⋯

(3) det ( \begin{matrix}x & −a_{2} \\ 1 & x+a_{1} \end{matrix} ) , det ( \begin{matrix}x & 0 & a_{3} \\ 1 & x & −a_{2} \\ 0 & 1 & x+a_{1} \end{matrix} ) , det ( \begin{matrix}x & 0 & 0 & −a_{4} \\ 1 & x & 0 & a_{3} \\ 0 & 1 & x & −a_{2} \\ 0 & 0 & 1 & x+a_{1} \end{matrix} ) , ⋯

3. Prove the following formula (the Vandermonde determinant):

det (\begin{matrix}1 & x_{1} & x_{1}^{2} & ⋯ x_{1}^{n−1} \\ ⋮ & & ⋮ & & ⋮ \\ 1 & x_{n} & x_{n}^{2} & ⋯ x_{n}^{n−1} \end{matrix}) = ∏_{i>j}(x_{i} − x_{j})

For example, for n = 2, 3 we have

det ( \begin{matrix}1 & x_{1} \\ 1 & x_{2} \end{matrix} ) = (x_{2} − x_{1}), det ( \begin{matrix}1 & x_{1} & x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} \\ 1 & x_{3} & x_{3}^{2} \end{matrix} ) = (x_{2} − x_{1})(x_{3} − x_{1})(x_{3} − x_{2}).

I encourage experimenting with the computer to answer question 2. Here is how you calculate determinants in Maple:

Input:  with(linalg):
Input:  A := matrix([[1,1,0],[1,1,1], [0,1,1]]);
Output:  A := matrix([[1, 1, 0], [1, 1, 1], [0, 1, 1]])
Input:  det(A);
Output:  -1
Input:  A:= matrix([[x,y,0],[1,z,1], [0,1,1]]);
Output:  A := matrix([[x, y, 0], [1, z, 1], [0, 1, 1]])
Input:  det(A);
Output:  x*z-x-y
Input:  B:= inverse(A);

Output: matrix([[(z-1)/(x*z-x-y), -y/(x*z-x-y), y/(x*z-x-y)], [-1/(x*z-x-y), x/(x*z-x-y), -x/(x*z-x-y)], [1/(x*z-x-y), -x/(x*z-x-y), (x*z-y)/(x*z-x-y)]])

Input:  det(B)
Output:  1/(x*z-x-y)

Bonus question - 15 points. Reed-Solomon Codes. Let 𝔽 be a finite field with q elements. List the non-zero elements of 𝔽_{q} as {β_{1}, …, β_{q−1}}. Define a map

𝔽_{q}[x]_{k} ⟶ 𝔽^{q−1},

by

f ↦ T (f) := (f(β_{1}), …, f(β_{q−1})).

Prove that T is a linear map and find when is it injective. When this holds, the image of T is a (n, k) code. Find the minimal Hamming weight of a non-zero element of the code. Compare your result with Assignment 2.