MathDB

IMC 1995 Problem 7

Source: IMC 1995

February 19, 2021

linear algebramatrixvector

Problem Statement

Let

A

be a

3\times 3

real matrix such that the vectors

Au

and

u

are orthogonal for every column vector

u\in \mathbb{R}^{3}

. Prove that: a)

A^{T}=-A

. b) there exists a vector

v \in \mathbb{R}^{3}

such that

Au=v\times u

for every

u\in \mathbb{R}^{3}

, where

v \times u

denotes the vector product in

\mathbb{R}^{3}

.

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