6.2. Revision: Determinants.

Before explaining how to find the eigenvalues of a linear transformation we need to review the notion of determinant. In a previous course you have met formulae for the determinants of 2x2 and 3x3 matrices:

(23)

and

(24)

This can be generalised to arbitrary square matrices; although we will not need the formulae in this course. What we will need are two basic properties of determinants. The first property is that the determinant is multiplicative.

Theorem 6.6. Let be matrices. Then

This has several important consequences. The first is that the identity matrix has unit determinant, since for all matrices . (Strictly speaking this uses the fact that there are some matrices with nonzero determinant.) Another consequence is that if is invertible, so that there exists with then

(25)

whence, in particular, . In fact, this is an equivalent characterisation of invertible matrices:

Theorem 6.7. Let be an matrix. Then is invertible if and only if .

One final consequence of Theorem 6.6 is that the determinant makes sense for a linear transformations in of an abstract vector space . To compute the determinant of we compute the determinant of the matrix of relative to a basis for . The choice of basis does not matter because if and are the matrices relative to two bases, then they are related by equation (20). Taking the determinant of both sides, using multiplicativity twice (Theorem 6.6) and using equation (25) for the change of basis matrix , we arrive at the fact that , which we define as .

A relatively efficient computational method to calculate the determinant of a square matrix is the following. First we bring to upper triangular form using only elementary row operations of types (2) and (3). Then

It follows from this method that the determinant of an matrix is an -multilinear function; in other words, if and , then

We can now go back to our discussion of eigenvalues.