8.6 A Formula For The Inverse
Note that this gives an easy way to write a formula for the inverse of an n × n matrix.
Theorem 8.6.1 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 8.5.3 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,
by Corollary 8.3.2 because there are two equal columns. Summarizing,
Using the other formula in Theorem 8.5.3, and similar reasoning,
This proves that if det
0, then A−1
exists with A−1
Now suppose A−1 exists. Then by Theorem 8.4.4,
The next corollary points out that if an n × n matrix A has a right or a left inverse, then it has an
Corollary 8.6.2 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 8.4.4 implies detB detA = 1 and so detA≠0. Therefore from Theorem
8.6.1, 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 8.6.1 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 exists,
thus solving the system. Now in the case that A−1 exists, there is a formula for A−1 given above. Using this
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 8.6.3 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 shown.
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 8.5.3.
Corollary 8.6.4 Let M be an upper (lower) triangular matrix. Then det
is obtained by taking
the product of the entries on the main diagonal.