Linear Algebra: A Modern Introduction

Published by Cengage Learning
ISBN 10: 1285463242
ISBN 13: 978-1-28546-324-7

Chapter 4 - Eigenvalues and Eigenvectors - 4.2 Determinants - Exercises 4.2 - Page 282: 55


$det(A)=0$ or $det(A)=1$

Work Step by Step

We know that if $A$ and $B$ are $n\times n$ matrices, then $det(AB)=det(A)det(B)$. If $A^2=A$, then $det(A^2)=det(A)\\det(A^2)-det(A)=0\\det(A)det(A)-det(A)=0\\det(A)(det(A)-1)=0$ Thus $det(A)=0$ or $det(A)=1$
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