Does a matrix have eigenvalues if its determinant is non-zero?

Question

I know that the determinant of ((lambda)(I)-(A) has to be zero, but what about det (A)? Does it have to be zero as well?

Answer 1

An n×n matrix will always have n eigenvalues counted with algebraic multiplicity. The eigenvalues are precisely those values of λ that make det (A-λI) = 0. Note that if det (A) = 0, then this is the same as saying det (A-0I) = 0, so det (A) will be zero if and only if 0 is an eigenvalue of the matrix.

Answer 2

a matrix has eigenvalues if it's determinant is non zero

sum of eigenvalues and their products are related to the trace of the matrix and it's determinant

trace of a matrix = sum of all diagonal entries of the matrix


try it and you can see that it is true:

Sum of eigenvalues = trace of the matrix.
Product of eigenvalues = determinant of the matrix

Answer 3

nope... the determinant of a matrix need not be zero for for it to have eigen values...