Show that 0 is a characteristic root of matrix A if and only if A is singular

source: adobe stock

 Solution:

 Let 0 be the characteristic equation of matrix A. Then implies ⌈A-ΥI⌋=0

 Hence  |A |=0 is singular. 

Conversely, let A is singular matrix, then

 | A-ΥI | = 0

⇒  |A |+(-Υ)|I| = 0 

⇒   (-Υ) 1 = 0 

implies, Υ = 0