Fundamentals of Matrix Algebra, 2011a

B 4,3 =
∣
1 2 8
−3 5 2
−1 9 6
∣
!
= ?
3.3 The Determinant
Again, we’re stuck. We won’t be able to fully compute C 4,3 ; all we know so far is that
C 4,3 = (−1) 4+3 B 4,3 = (−1)B 4,3 .
Once we learn how to compute determinates for matrices larger than 2 × 2 we can
come back and finish this exercise. .
In our previous example we ran into a bit of trouble. By our definion, in order
to compute a minor of an n × n matrix we needed to compute the determinant of a
(n − 1) × (n − 1) matrix. This was fine when we started with a 3 × 3 matrix, but when
we got up to a 4 × 4 matrix (and larger) we run into trouble.
We are almost ready to define the determinant for any square matrix; we need
one last definion.
.
Ḋefinion 25
Cofactor Expansion
Let A be an n × n matrix.
.
The cofactor expansion of A along the i th row is the sum
a i,1 C i,1 + a i,2 C i,2 + · · · + a i,n C i,n .
The cofactor expansion of A down the j th column is the sum
a 1,j C 1,j + a 2,j C 2,j + · · · + a n,j C n,j .
The notaon of this definion might be a lile inmidang, so let’s look at an example.
. Example 70 .Let
⎡
A = ⎣ 1 2 3 ⎤
4 5 6 ⎦ .
7 8 9
Find the cofactor expansions along the second row and down the first column.
S
the sum
By the definion, the cofactor expansion along the second row is
a 2,1 C 2,1 + a 2,2 C 2,2 + a 2,3 C 2,3 .
(Be sure to compare the above line to the definion of cofactor expansion, and see
how the “i” in the definion is replaced by “2” here.)
139