CHAPTER 10.09: EIGENVALUES AND EIGENVECTORS: Theorems of eigenvalues and eigenvectors Part 2 of 6     In this segment we will talk about some theorems of eigenvalues and eigenvectors so the second theorem right here saying that if λ=0 is an eigenvalue of a nxn [A] matrix, then [A] is noninvertible so if you find out that one of the eigenvalues of the [A] matrix of a square matrix turns out to be zero just one then the [A] matrix is noninvertible which is the same as saying (singular) which is the same as saying its inverse doesn’t exist and so on and so forth so that’s what it is telling you so for example if somebody says that hey this is my matrix given to me [A]=5, 6, 2, 3, 5, 9, 2, 1, -7 for this particular matrix if you go through the process of finding the eigenvalues you will find that λ=0 is an eigenvalue youll find one of the eigenvalues is zero so what does that mean that means that [A] inverse does not exist that means that A inverse does not exist or it means that [A] is not invertible it means [A] is singular it also means that the det(A)=0 because if A inverse does not exist then we know that for a square matrix the det of that matrix has to be equal to zero that’s the end of this segment