A First Course in Linear Algebra, 2017a

Chapter 3
Determinants
3.1 Basic Techniques and Properties
Outcomes
A. Evaluate the determinant of a square matrix using either Laplace Expansion or row operations.
B. Demonstrate the effects that row operations have on determinants.
C. Verify the following:
(a) The determinant of a product of matrices is the product of the determinants.
(b) The determinant of a matrix is equal to the determinant of its transpose.
3.1.1 Cofactors and 2 × 2 Determinants
Let A be an n × n matrix. That is, let A be a square matrix. The determinant of A, denoted by det(A) is a
very important number which we will explore throughout this section.
If A is a 2×2 matrix, the determinant is given by the following formula.
Definition 3.1: Determinant of a Two By Two Matrix
[ ]
a b
Let A = . Then
c d
det(A)=ad − cb
The determinant is also often denoted by enclosing the matrix with two vertical lines. Thus
[ ]
a b
det =
c d ∣ a b
c d ∣ = ad − bc
The following is an example of finding the determinant of a 2 × 2matrix.
Example 3.2: A Two by Two Determinant
[ ]
2 4
Find det(A) for the matrix A = .
−1 6
Solution. From Definition 3.1,
det(A)=(2)(6) − (−1)(4)=12 + 4 = 16
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