A First Course in Linear Algebra, 2017a

112 Determinants
The following provides a formal definition for the determinant of an n × n matrix. You may wish
to take a moment and consider the above definitions for 2 × 2and3× 3 determinants in context of this
definition.
Definition 3.11: The Determinant of an n × n Matrix
Let A be an n × n matrix where n ≥ 2 and suppose the determinant of an (n − 1) × (n − 1) has been
defined. Then
n
n
det(A)= ∑ a ij cof(A) ij = ∑ a ij cof(A) ij
j=1
i=1
The first formula consists of expanding the determinant along the i th row and the second expands
the determinant along the j th column.
In the following sections, we will explore some important properties and characteristics of the determinant.
3.1.2 The Determinant of a Triangular Matrix
There is a certain type of matrix for which finding the determinant is a very simple procedure. Consider
the following definition.
Definition 3.12: Triangular Matrices
AmatrixA is upper triangular if a ij = 0 whenever i > j. Thus the entries of such a matrix below
the main diagonal equal 0, as shown. Here, ∗ refers to any nonzero number.
⎡
⎤
∗ ∗ ··· ∗
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.
The following theorem provides a useful way to calculate the determinant of a triangular matrix.
Theorem 3.13: Determinant of a Triangular Matrix
Let A be an upper or lower triangular matrix. Then det(A) is obtained by taking the product of the
entries on the main diagonal.
The verification of this Theorem can be done by computing the determinant using Laplace Expansion
along the first row or column.
Consider the following example.