Page 96 - A Course in Linear Algebra with Applications

80 Chapter Three: Determinants

Now we can see from the form of the elementary matrix E
that det(E) equals — 1, 1 or c, respectively, in the three cases;
hence the formula det(AE ) = det(A) det(E) is valid. In short
our formula is true when B is an elementary matrix. Applying
this fact repeatedly, we find that &et(AB) equals
• • • E 2E X) = det(A) det(E k) • • • det(£ 2 ) det(Ei),
det(AE fc
which shows that

det(AB) = det(A) det{E k • • • E x) = det(A) det(5).

Corollary 3.3.4
Let A and B be n x n matrices. If AB = I n , then BA = I n,
1
and thus B = A" .
Proof
For 1 = det(AB) = det(A) det(S), so det(A) ^ 0 and A is in-
l
l
vertible, by 3.3.1. Therefore BA = A~ {AB)A = A~ I nA =
In-
Corollary 3.3.5
-1
If A is an invertible matrix, then det(^4 ) = l/det(A).
Proof
1
l
Clearly 1 = det(7) = det{AA~ ) = det(A)det(A' ), from
which the statement follows.
The adjoint matrix
Let A = (a,ij) be an n x n matrix. Then the adjoint
matrix
adj (A)
of A is defined to be the nxn matrix whose (i,j) element is the
(j,i) cofactor Aji of A. Thus adj(yl) is the transposed matrix
of cofactors of A. For example, the adjoint of the matrix

(6 -1 3 ]
\2 -3 4/