Linear Algebra, Theory And Applications, 2012a

78 DETERMINANTS
Note that for a n × n matrix, you will need n! terms to evaluate the determinant in this
way. If n =10, this is 10! = 3, 628 , 800 terms. This is a lot of terms.
In addition to the difficulties just discussed, why is the determinant well defined? Why
should you get the same thing when you expand along any row or column? I think you
should regard this claim that you always get the same answer by picking any row or column
with considerable skepticism. It is incredible and not at all obvious. However, it requires
a little effort to establish it. This is done in the section on the theory of the determinant
which follows.
Notwithstanding the difficulties involved in using the method of Laplace expansion,
certain types of matrices are very easy to deal with.
Definition 3.1.5 A matrix M, is upper triangular if M ij =0whenever i>j. Thus such
a matrix equals zero below the main diagonal, the entries of the form M ii , as shown.
⎛
⎞
∗ ∗ ··· ∗
. 0 ∗ .. . ⎜
⎝
.
. .. . ⎟ .. ∗ ⎠
0 ··· 0 ∗
A lower triangular matrix is defined similarly as a matrix for which all entries above the
main diagonal are equal to zero.
You should verify the following using the above theorem on Laplace expansion.
Corollary 3.1.6 Let M be an upper (lower) triangular matrix. Then det (M) is obtained
by taking the product of the entries on the main diagonal.
Proof: The corollary is true if the matrix is one to one. Suppose it is n × n. Then the
matrix is of the form (
m11 a
0 M 1
)
where M 1 is (n − 1)×(n − 1) . Then expanding along the first row, you get m 11 det (M 1 )+0.
Then use the induction hypothesis to obtain that det (M 1 )= ∏ n
i=2 m ii.
Example 3.1.7 Let
Find det (A) .
A =
⎛
⎜
⎝
1 2 3 77
0 2 6 7
0 0 3 33.7
0 0 0 −1
⎞
⎟
⎠
From the above corollary, this is −6.
There are many properties satisfied by determinants. Some of the most important are
listed in the following theorem.
Theorem 3.1.8 If two rows or two columns in an n × n matrix A are switched, the determinant
of the resulting matrix equals (−1) times the determinant of the original matrix. If
A is an n×n matrix in which two rows are equal or two columns are equal then det (A) =0.
Suppose the i th row of A equals (xa 1 + yb 1 , ··· ,xa n + yb n ).Then
det (A) =x det (A 1 )+y det (A 2 )