A.9.6 A Formula For The Inverse
Note that this gives an easy way to write a formula for the inverse of an n × n
matrix. Recall the definition of the inverse of a matrix in Definition A.6.7 on Page
Theorem A.9.18 A−1 exists if and only if det(A)≠0. If det(A)≠0, then
ij the ijth cofactor of A.
Proof: By Theorem A.9.17 and letting
, if det
Now in the matrix A, replace the kth column with the rth column and then expand along the
kth column. This yields for k≠r,
because there are two equal columns by Corollary A.9.9. Summarizing,
Using the other formula in Theorem A.9.17, 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.9.13,
The next corollary points out that if an n×n matrix A has a right or a left inverse, then
it has an inverse.
Corollary A.9.19 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.9.13 implies detB detA = 1 and so detA≠0. Therefore
from Theorem A.9.18, 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.9.18 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 of
In case you are solving a system of equations, Ax = y for x, it follows that if A−1
thus solving the system. Now in the case that A−1 exists, there is a formula for A−1 given
above. Using this formula,
By the formula for the expansion of a determinant along a column,
where here the ith column of A is replaced with the column vector,
, and the
determinant of this modified matrix is taken and divided by det
. This formula is known
as Cramer’s rule.
Definition A.9.20 A matrix M, is upper triangular if Mij = 0 whenever i > j.
Thus such a matrix equals zero below the main diagonal, the entries of the form Mii as
A lower triangular matrix is defined similarly as a matrix for which all entries above the main
diagonal are equal to zero.
With this definition, here is a simple corollary of Theorem A.9.17.
Corollary A.9.21 Let M be an upper (lower) triangular matrix. Then det
is obtained by taking the product of the entries on the main diagonal.