Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 167: 17


$W$ is not a vector subspace of $M_{n,n}$.

Work Step by Step

Let $u\in W$ and $c=2\in R$. Now, $$\det(u)=1, \quad \det(cu)=c\det(u)=2$$ which means that $cu$ is not an element of $W$. Hence, $W$ is not a vector subspace of $M_{n,n}$.
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