Page 244 - Linear Algebra Done Right

Determinant of a Matrix
The theorem below states that the determinant of an operator equals
the determinant of the matrix of the operator. This theorem does not
specify a basis because, by the corollary above, the determinant of the 235
matrix of an operator is the same for every choice of basis.
10.33 Theorem: If T ∈L(V), then det T = det M(T).
Proof: Let T ∈L(V). As noted above, 10.32 implies that det M(T)
is independent of which basis of V we choose. Thus to show that

det T = det M(T)

for every basis of V, we need only show that the equation above holds
for some basis of V. We already did this (on page 230), choosing a basis
of V with respect to which M(T) is an upper-triangular matrix (if V is a
complex vector space) or an appropriate block upper-triangular matrix
(if V is a real vector space).

If we know the matrix of an operator on a complex vector space, the
theorem above allows us to find the product of all the eigenvalues with-
out finding any of the eigenvalues. For example, consider the operator
5
on C whose matrix is
 
0 0 0 0 −3
 
 1 0 0 0 6 
 
 0 1 0 0 0  .
 


 0 0 1 0 0 
0 0 0 1 0
No one knows an exact formula for any of the eigenvalues of this opera-
tor. However, we do know that the product of the eigenvalues equals −3
because the determinant of the matrix above equals −3.
The theorem above also allows us easily to prove some useful prop-
erties about determinants of operators by shifting to the language of
determinants of matrices, where certain properties have already been
proved or are obvious. We carry out this procedure in the next corol-
lary.

10.34 Corollary: If S, T ∈L(V), then

det(ST) = det(TS) = (det S)(det T).