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Chapter 10. Trace and Determinant
230
Let’s compute the determinant of an upper-triangular matrix
a 1,1 . ∗
A = . . .
0 a n,n
The permutation (1, 2,...,n) has sign 1 and thus contributes a term
of a 1,1 ...a n,n to the sum 10.25 defining det A. Any other permutation
(m 1 ,...,m n ) ∈ perm n contains at least one entry m j with m j >j,
which means that a m j ,j = 0 (because A is upper triangular). Thus all
the other terms in the sum 10.25 defining det A make no contribu-
tion. Hence det A = a 1,1 ...a n,n . In other words, the determinant of an
upper-triangular matrix equals the product of the diagonal entries. In
particular, this means that if V is a complex vector space, T ∈L(V),
and we choose a basis of V with respect to which M(T) is upper trian-
gular, then det T = det M(T). Our goal is to prove that this holds for
every basis of V, not just bases that give upper-triangular matrices.
Generalizing the computation from the paragraph above, next we
will show that if A is a block upper-triangular matrix
A 1 ∗
.
A = . . ,
0 A m
where each A j is a 1-by-1 or 2-by-2 matrix, then
10.27 det A = (det A 1 )...(det A m ).
To prove this, consider an element of perm n. If this permutation
moves an index corresponding to a 1-by-1 block on the diagonal any-
place else, then the permutation makes no contribution to the sum
10.25 defining det A (because A is block upper triangular). For a pair
of indices corresponding to a 2-by-2 block on the diagonal, the permu-
tation must either leave these indices fixed or interchange them; oth-
erwise again the permutation makes no contribution to the sum 10.25
defining det A (because A is block upper triangular). These observa-
tions, along with the formula 10.26 for the determinant of a 2-by-2 ma-
trix, lead to 10.27. In particular, if V is a real vector space, T ∈L(V),
and we choose a basis of V with respect to which M(T) is a block
upper-triangular matrix with 1-by-1 and 2-by-2 blocks on the diagonal
as in 9.9, then det T = det M(T).
