Linear Algebra and Its Applications · Problem 1.2.22

Matrices and Gaussian Elimination — Gilbert Strang

Mathematics · geometry-of-linear-equations

Problem

(a,b)

is a multiple of

(c,d)

with

abcd \neq 0

, show that

(a,c)

is a multiple of

(b,d)

. This is surprisingly important: call it a challenge question. You could use numbers first to see how

a

b

c

, and

d

are related. The question will lead to: If

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

has dependent rows then it has dependent columns.