In Section ?? we introduced determinants for matrices . There we showed that the determinant of is nonzero if and only if is invertible. In Section ?? we saw that the eigenvalues of are the roots of its characteristic polynomial, and that its characteristic polynomial is just the determinant of a related matrix, namely, .
In Section ?? we generalize the concept of determinants to matrices, and in Section ?? we use determinants to show that every matrix has exactly eigenvalues — the roots of its characteristic polynomial. Properties of eigenvalues are also discussed in detail in Section ??. Certain details concerning determinants are deferred to Appendix ??.