Page 97 - A Course in Linear Algebra with Applications

3.3: Determinants and Inverses of Matrices 81

is
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The significance of the adjoint matrix is made clear by
the next two results.
Theorem 3.3.6
// A is any n x n matrix, then

A adj(A) = (det(A))I n = adj(A)A.

Proof
The (i,j) entry of the matrix product A adj(i4) is

n n
^2aik(adj(A))kj = ^2a ikA jk.
k=i fc=i

If i = j , this is just the expansion of det(A) by rov/ i; on the
other hand, if i ^ j , the sum is also a row expansion of a
determinant, but one in which rows i and j are identical. By
3.2.2 the sum will vanish in this case. This means that the
off-diagonal entries of the matrix product A a,d](A) are zero,
while the entries on the diagonal all equal det(^4). Therefore
A adj(^4) is the scalar matrix (det(A))I n, as claimed. The
second statement can be proved in a similar fashion.

Theorem 3.3.5 leads to an attractive formula for the in-
verse of an invertible matrix.
Theorem 3.3.7
If A is an invertible matrix, then A" 1 = (l/det(A))adj(A).