A First Course in Linear Algebra, 2017a

114 Determinants
3.1.3 Properties of Determinants I: Examples
There are many important properties of determinants. Since many of these properties involve the row
operations discussed in Chapter 1, we recall that definition now.
Definition 3.15: Row Operations
The row operations consist of the following
1. Switch two rows.
2. Multiply a row by a nonzero number.
3. Replace a row by a multiple of another row added to itself.
We will now consider the effect of row operations on the determinant of a matrix. In future sections,
we will see that using the following properties can greatly assist in finding determinants. This section will
use the theorems as motivation to provide various examples of the usefulness of the properties.
The first theorem explains the affect on the determinant of a matrix when two rows are switched.
Theorem 3.16: Switching Rows
Let A be an n × n matrix and let B be a matrix which results from switching two rows of A. Then
det(B)=−det(A).
When we switch two rows of a matrix, the determinant is multiplied by −1. Consider the following
example.
Example 3.17: Switching Two Rows
[ ]
[ ]
1 2
3 4
Let A = and let B = . Knowing that det(A)=−2, finddet(B).
3 4
1 2
Solution. By Definition 3.1,det(A)=1 × 4 − 3 × 2 = −2. Notice that the rows of B are the rows of A but
switched. By Theorem 3.16 since two rows of A have been switched, det(B)=−det(A)=−(−2)=2.
You can verify this using Definition 3.1.
♠
The next theorem demonstrates the effect on the determinant of a matrix when we multiply a row by a
scalar.
Theorem 3.18: Multiplying a Row by a Scalar
Let A be an n × n matrix and let B be a matrix which results from multiplying some row of A by a
scalar k. Thendet(B)=k det(A).