Elementary Linear Algebra 7th Edition

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

Chapter 2 - Matrices - 2.1 Operations with Matrices - 2.1 Exercises - Page 50: 72

Answer

There do not exist matrices $A,B$ such that $AB-BA=I$

Work Step by Step

Since $A$ and $B$ are square matrices of order $2$, $\therefore Tr(AB)=tr(BA)$ If $AB-BA=I$ (where $I$ represents the identity matrix) is true, then $Tr(AB-BA)=Tr(I)\\ Tr(AB)-Tr(BA)=Tr(I)\\ $ but the $LHS =0$ and $RHS=1$. Hence, there do not exist matrices $A,B$ such that $AB-BA=I$
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