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: 32


The set $W$ is a vector subspace $M_{n,n}$.

Work Step by Step

The set $W$ of all $n \times n$ matrices that commute with a given matrix $B $ is a vector subspace $M_{n,n}$. Indeed; Let $A$ and $C$ be two matrices that commute with the matrix $B$. Now, $(A+C)B=AB+AC=BA+BC=B(A+C)$ and also for any $c\in R$ $(cA)B=B(cA)$, hence $W$ is a vector subspace $M_{n,n}$.
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