19.4.1 A Formula For The Inverse
The definition of the determinant in terms of Laplace expansion along a row or column
also provides a way to give a formula for the inverse of a matrix. Recall the definition of
the inverse of a matrix in Definition 18.6.2 on Page 1157. Also recall the definition of the
cofactor matrix given in Definition 19.2.12 on Page 1241. This cofactor matrix was just
the matrix which results from replacing the ijth entry of the matrix with the ijth
The following theorem 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.
In other words, A−1 is equal to one divided by the determinant of A times
the adjugate matrix of A. This is what the following theorem says with more
precision. The proof is presented later in the appendix devoted to the theory of the
Theorem 19.4.1 A−1 exists if and only if det(A)≠0. If det(A)≠0, then
ij the ijth cofactor of A.
Example 19.4.2 Find the inverse of the matrix
First find the determinant of this matrix. Using Theorems 19.2.23 - 19.2.25
on Page 1247, the determinant of this matrix equals the determinant of the
which equals 12. The cofactor matrix of A is
Each entry of A was replaced by its cofactor. Therefore, from the above theorem, the
inverse of A should equal
Does it work? You should check to see if it does. When the matrices are multiplied
and so it is correct.
Example 19.4.3 Find the inverse of the matrix
First find its determinant. This determinant is
The inverse is therefore equal
Expanding all the 2 × 2 determinants this yields
Always check your work.
and so we got it right. If the result of multiplying these matrices had been something
other than the identity matrix, you would know there was an error. When this happens,
you need to search for the mistake if you are interested in getting the right
answer. A common mistake is to forget to take the transpose of the cofactor