Inverses and the Adjoint Matrix
We now use Cramer’s rule to develop an explicit formula for the inverse A−1 of the invertible matrix A. First, we need to rewrite Cramer’s rule more concisely. Expansion of the determinant in the numerator in Eq. (24) along its i th column yields
xi=1|A|(b1A1i+b2A2i+⋯+bnAni),
(25)
because the cofactor of bp is simply the cofactor Api of api in |A|. The formula in Eq. (25) gives the solution vector
Then the definition of matrix multiplication yields
x=1|A|⎡⎣⎢⎢⎢⎢⎢A11A12⋮A1nA21A22⋮A2n⋯⋯⋱⋯An1An2⋮Ann⎤⎦⎥⎥⎥⎥⎥⎡⎣⎢⎢⎢⎢b1b2⋮bn⎤⎦⎥⎥⎥⎥
(26)
for the solution x of Ax=b.
Observe that the double subscripts in (26) are reversed from their usual order; the element in the ith row and jth column is Aji (rather than Aij). We therefore see in (26) the transpose of the cofactor matrix [Aij] of then n×n matrix A. The transpose of the cofactor matrix of A is called the adjoint matrix of A and is denoted by
adjA=[Aij]T=[Aij].
(27)
With the aid of this notation, Cramer’s rule as expressed in Eq. (26) can be written in the especially simple form
x=[Aji]b|A|=(adjA)b|A|.
(28)
The fact that the formula in (28) gives the unique solution x of Ax=b implies that
A(adjA)b|A|=b
(29)
for every n-vector b. If we write
for brevity, then
for every n-vector b. From this it follows (one column at a time, as we use Fact 2 in Section 3.5) that
for every matrix B having n rows. In particular, with B=I (the n×n identity matrix), we see that
Therefore, we have discovered that the matrix C as defined in Eq. (30) is the inverse matrix A−1 of A.