A.5.10 Formula for the Inverse
Note that this gives an easy way to write a formula for the inverse of an n × n matrix.
Theorem A.5.17 A−1 exists if and only if det(A)≠0. If det(A)≠0, then A−1 =
ij the ijth cofactor of A.
Proof: By Theorem A.5.16 and letting
, if det
when k≠r. Replace the kth column with the rth column to obtain a matrix Bk whose determinant equals
zero by Corollary A.5.9. However, expanding this matrix along the kth column yields
Using the other formula in Theorem A.5.16, and similar reasoning,
This proves that if det
0, then A−1
exists with A−1
Now suppose A−1 exists. Then by Theorem A.5.13,
The next corollary points out that if an n × n matrix A has a right or a left inverse, then it has an
Corollary A.5.18 Let A be an n×n matrix and suppose there exists an n×n matrix B such that
BA = I. Then A−1 exists and A−1 = B. Also, if there exists C an n × n matrix such that AC = I,
then A−1 exists and A−1 = C.
Proof: Since BA = I, Theorem A.5.13 implies
and so detA≠0. Therefore from Theorem A.5.17, A−1 exists. Therefore,
The case where CA = I is handled similarly. ■
The conclusion of this corollary is that left inverses, right inverses, and inverses are all the same in the
context of n × n matrices.
Theorem A.5.17 says that to find the inverse, take the transpose of the cofactor matrix and divide by
the determinant. The transpose of the cofactor matrix is called the adjugate or sometimes the classical
adjoint of the matrix A. It is an abomination to call it the adjoint although you do sometimes see it
referred to in this way. In words, A−1 is equal to one over the determinant of A times the adjugate matrix