MATH 110: LINEAR ALGEBRA HOMEWORK #6 WORKSHEET

JOHN VOIGHT

Choose a partner. Between the two of you, choose a problem you will both try to solve. Work on it individually, and then when you have made sufficient progress share your work with your partner. Write together with your partner a nice solution. Then find another pair and see if you can explain your work to them. (When you hear their solution, be picky!)

(1) Label true or false, and explain.

(a) For any m × n-matrix A with rank r, there exists an invertible m × m matrix P and an invertible n × n matrix Q such that

A = P (\begin{matrix}I_{r} & O \\ O & O \end{matrix}) Q

where I_{r} is the r × r identity matrix.

(b) For any n × n-matrix A with rank r, there exists an invertible n × n matrix P such that

A = P (\begin{matrix}I_{r} & O \\ O & O \end{matrix}) P^{−1}.

(c) If A, B are n × n matrices over a field F, then rk(A) = rk(B) implies rk(A^{2}) = rk(B^{2}).

(d) rk(A + B) ≤ rk(A) + rk(B).

(e) Let A ∈ M_{n×n}(ℝ) be a real n × n-matrix. Suppose there exists a B ∈ M_{n×n}(ℂ) with complex coefficients such that AB = I. Then B actually has real coefficients.

(2) Let A be an n × n matrix with entries 1 and −1. Show that the integer det(A) is divisible by 2^{n−1}.

(3) Let f(x) = (p_{1} − x)(p_{2} − x) ⋯ (p_{n} − x) ∈ ℝ[x] and let Δ_{n} be the determinant of the matrix

(\begin{matrix}p_{1} & a & a & … & a \\ b & p_{2} & a & … & a \\ b & b & p_{3} & … & a \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ b & b & b & … & p_{n} \end{matrix})

where a, b ∈ ℝ. Show that if a ≠ b, that

Δ_{n} = \frac{bf(a) − af(b)}{b − a}.

Conjecture a formula when a = b.

Date: March 2, 2005.