Page 95 - A Course in Linear Algebra with Applications

3.3: Determinants and Inverses of Matrices 79

Corollary 3.3.2
A linear system AX = 0 with n equations in n unknowns has
a non-trivial solution if and only if det(A) = 0.
This very useful result follows directly from 2.3.5 and
3.3.1. Theorem 3.3.1 can be used to establish a basic formula
for the determinant of the product of two matrices.
Theorem 3.3.3
If A and B are any n x n matrices, then

det(AB) = det (A) det(J5).

Proof
Consider first the case where B is not invertible, which by
3.3.1 means that det(B) = 0. According to 2.3.5 there is a
non-zero vector X such that BX = 0. This clearly implies
that (AB)X = 0, and so, by 2.3.5 and 3.3.1, det(AJ3) must
also be zero. Thus the formula certainly holds in this case.
Suppose now that B is invertible. Then B is a product
of elementary matrices, say B = E\E% • • • Ek', this is by 2.3.5.
Now the effect of right multiplication of A by an elementary
matrix E is to apply an elementary column operation to A.
What is more, we can tell from 3.2.3 just what the value of
det(AE) is; indeed

{ -det(A) ,
det (A)

c det (A)
according to whether E represents a column operation of the
types

O i 4—^ O j
Ci + cCj
cCi