Linear Algebra Notes
Chapter 3
DETERMINANT AND TRACE
a
We have seen the determinant of a matrix A =
c
b
:
d
det A = ad − bc.
This chapter describes arithmetic properties of the determinant, and its cousin, the
trace (defined below).
Multiplicative Property of Determinant. If A and B are matrices, then
det(AB) = det(A) det(B).
Proof. We are given two arbitrary matrices
a b
A=
,
and
c d
Then
a0
B= 0
c
aa0 + bc0
AB =
ca0 + dc0
b0
.
d0
ab0 + bd0
,
cb0 + dd0
so
det(AB) = (aa0 + bc0 )(cb0 + dd0 ) − (ab0 + bd0 )(ca0 + dc0 )
= aa0 dd0 + bc0 cb0 − ab0 dc0 − bd0 ca0
= (ad − bc)(a0 d0 − b0 c0 )
= det(A) det(B).
Inverse Property of Determinant. If A
−1
exists then det(A) is nonzero and
1
.
det(A−1 ) =
det A
Proof. By definition we have AA−1 = I, so
det(AA−1 ) = det(I) = 1.
By the multiplicative property we have
det(AA−1 ) = det(A) det(A−1 ).
Therefore
det(A−1 ) det(A) = 1,
so
det(A−1 ) =
as claimed.
1
1
,
det(A)
2
Conjugation Property of Determinant. If A is any matrix and B is any invertible matrix then
det(B −1 AB) = det(A).
Proof. Using the multiplicative property (twice) we have
det(B −1 AB) = det(B −1 ) det(A) det(B).
The right side of this equation is a product of three numbers, which commute, so
det(B −1 AB) = det(B −1 ) det(B) det(A) = det(A),
by the inverse property.
The trace of a matrix is the sum of the diagonal entries.
a b
tr
= a + d.
c d
The multiplicative property of the trace is weaker than that for the determinant.
Multiplicative Property of Trace. For any two matrices A and B we have
tr(AB) = tr(BA).
Proof. See exercises. There is no inverse property for the trace, but the conjugation property is exactly
the same:
Conjugation Property of Trace. If A is any matrix and B is any invertible
matrix then
tr(B −1 AB) = tr(A).
Proof. Using the multiplicative property of trace we have
tr(B −1 AB) = tr(BB −1 A) = tr(IA) = tr(A).
Some other formulas (and non-formulas) for the trace and determinant appear
in the exercise below.
Both the determinant and trace have a geometrical meaning, which we will see
later. We will also combine the trace and determinant to make the characteristic
polynomial, which contains all the essential features of a matrix. (The entries in
the matrix are not its essential features!)
Exercise 3.1. Prove or disprove: 1
(a) tr(AB) = tr(A) tr(B).
(b) tr(AB) = tr(BA).
(c) tr(A + B) = tr(A) + tr(B).
1
(d) tr(A−1 ) = tr(A)
.
(e) tr(xA) = x tr(A). (Here x is a scalar.)
(f) det(xA) = x det(A).
(g) det(A + B) = det(A) + det(B).
(h) det(xI − A) = x2 − tr(A)x + det(A).
1 That is, if the formula is true for all matrices A, B, then derive the formula. If the formula is
false for some A, B then give an example of such matrices where the formula is false.