Higher Algebra 2 — Spring 2006

Gerald Hoehn

Problem sheet 2 February 3, 2006

Problem 1: Let x_{1}, …, x_{n} be elements of a field K. Prove the following formula for the Vandermonde determinant:

| \begin{matrix}1 & x_{1} & … & x_{1}^{n−1} \\ 1 & x_{2} & … & x_{2}^{n−1} \\ ⋮ & ⋮ & & ⋮ \\ 1 & x_{n} & … & x_{n}^{n−1} \end{matrix} | = ∏_{i<j}(x_{j} − x_{i}).

Problem 2: Prove the following theorem about the rank of a matrix: Let A ∈ Mat(n × m, K) and let r be the largest non-negative integer such that there exist an r × r submatrix A′ of A with det A′ ≠ 0. Then rank A = r. Here, a submatrix of A is a matrix obtained by removing an abitrary number of rows and columns from A.

Problem 3: Suppose A is an n × n matrix with real entries such that the diagonal elements are all positive, the off-diagonal elements are all negative and the row sums are all positive. Prove that det A ≠ 0.

Problem 4: Compute the determinant of the matrix

A = ( \begin{matrix}B_{1} & C \\ 0 & B_{2} \end{matrix} )

in terms of the three submatrices B_{1}, B_{2} and C.

Problem 5^{∗}: Let A be an n × n matrix. Let I = {i_{1}, …, i_{p}} be a subset of {1, …, n} and let J = {j_{1}, …, j_{q}} be the complement of I in {1, …, n}. Denote by Γ the set of all permutations σ ∈ S_{n} such that the restrictions of the map σ to I and J are monotone increasing functions. For γ ∈ Γ let A_{γ} be the submatrix of A obtained by removing the rows j_{1}, …, j_{q} and columns γ(j_{1}), …, γ(j_{q}) from A and let A_{γ}^{∗} be the submatrix of A obtained by removing the rows i_{1}, …, i_{p} and columns γ(i_{1}), …, γ(i_{p}) from A. Prove the following expansion theorem of Laplace:

det A = ∑_{γ∈Γ} ϵ(γ) · det A_{γ} · det A_{γ}^{∗}.