A First Course in Linear Algebra, 2017a

3.1. Basic Techniques and Properties 109
Definition 3.5: The ij th Cofactor of a Matrix
Suppose A is an n × n matrix. The ij th cofactor, denoted by cof(A) ij is defined to be
cof(A) ij =(−1) i+ j minor(A) ij
It is also convenient to refer to the cofactor of an entry of a matrix as follows. If a ij is the ij th entry of
the matrix, then its cofactor is just cof(A) ij .
Example 3.6: Finding Cofactors of a Matrix
Consider the matrix
Find cof(A) 12 and cof(A) 23 .
⎡
A = ⎣
1 2 3
4 3 2
3 2 1
⎤
⎦
Solution. We will use Definition 3.5 to compute these cofactors.
First, we will compute cof(A) 12 . Therefore, we need to find minor(A) 12 . This is the determinant of
the 2 × 2 matrix which results when you delete the first row and the second column. Thus minor(A) 12 is
given by
[ ]
4 2
det = −2
3 1
Then,
cof(A) 12 =(−1) 1+2 minor(A) 12 =(−1) 1+2 (−2)=2
Hence, cof(A) 12 = 2.
Similarly, we can find cof(A) 23 . First, find minor(A) 23 , which is the determinant of the 2 × 2matrix
which results when you delete the second row and the third column. This minor is therefore
[ ] 1 2
det = −4
3 2
Hence,
cof(A) 23 =(−1) 2+3 minor(A) 23 =(−1) 2+3 (−4)=4
♠
You may wish to find the remaining cofactors for the above matrix. Remember that there is a cofactor
for every entry in the matrix.
We have now established the tools we need to find the determinant of a 3 × 3matrix.