21.2.1 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
5.4.2 on Page 257. Also recall the definition of the cofactor matrix given in Definition 21.1.12 on Page 1033.
This cofactor matrix was just the matrix which results from replacing the ijth entry of the matrix with the
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.
Theorem 21.2.1 A−1 exists if and only if det(A)≠0. If det(A)≠0, then A−1 =
ij the ijth cofactor of A.
Example 21.2.2 Find the inverse of the matrix
First find the determinant of this matrix. Using Theorems 21.1.23 - 21.1.25 on Page 1038, the
determinant of this matrix equals the determinant of the matrix
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
Does it work? You should check to see if it does. When the matrices are multiplied
and so it is correct.
Example 21.2.3 Find the inverse of the matrix
First find its determinant. This determinant is
The inverse is therefore equal to
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 matrix.
Proof of Theorem 21.2.1: From the definition of the determinant in terms of expansion along a
column, 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 Theorem 21.1.21. However, expanding this matrix Bk along the kth column
which is the krth entry of cof
Using the other formula in Definition 21.1.13, and similar reasoning,
which is the rkth entry of Acof
and it follows from 21.2 and 21.3 that A−1 =
In other words,
Now suppose A−1 exists. Then by Theorem 21.1.26,
This way of finding inverses is especially useful in the case where it is desired to find the inverse of a
matrix whose entries are functions.
Example 21.2.4 Suppose
Show that A
−1 exists and then find it.
First note det
0 so A
exists. The cofactor matrix is
and so the inverse is