Linear algebra is a branch of mathematics that deals with the study of vector spaces and linear mappings between them. It provides a framework for solving systems of linear equations and analyzing geometric objects such as lines, planes, and transformations in multiple dimensions.
• Section 2.1: Determinants by Cofactor Expansion.
• Section 2.2: Evaluating Determinants by Row Reduction.
• Section 2.3: Properties of Determinants; Cramer’s Rule.
, |Math 231|Section 2.1|Page 2
Section 2.1: Determinants by Cofactor Expansion1
Concepts:
• Determinant
• Minor
• Cofactor
• Cofactor expansion
Learning Outcomes.
After completing this section, you should be able to:
• Find the minors and cofactors of a square matrix.
• Use cofactor expansion to evaluate the determinant of a square matrix.
• Use the arrow technique to evaluate the determinant of a 2 × 2 or 3 × 3
matrix.
• Use the determinant of a 2 × 2 invertible matrix to find the inverse of that
matrix.
• Find the determinant of an upper triangular, lower triangular, or diagonal
matrix by inspection.
1
The materials of these lecture notes are based on the textbook of the course.
, |Math 231|Section 2.1|Page 3
Determinants
The determinant of a square matrix 𝐴𝐴 is a single real number which contains an
important amount of information about the matrix 𝐴𝐴. We denote the determinant of
the matrix 𝐴𝐴 by det(𝐴𝐴) or |𝐴𝐴|.
Determinants are defined only for square matrices. If a matrix can be exhibited, we
denote its determinant by replacing the brackets with vertical straight lines.
𝟏𝟏 × 𝟏𝟏 Matrices
If 𝐴𝐴 = [𝑎𝑎]1×1 matrix, then the determinant of 𝐴𝐴 is defined by
det(𝐴𝐴) = |𝐴𝐴| = 𝑎𝑎
𝟐𝟐 × 𝟐𝟐 Matrices
If 𝐴𝐴 is the 2 × 2 matrix
𝑎𝑎11 𝑎𝑎12
𝐴𝐴 = �𝑎𝑎 𝑎𝑎22 �,
21
then the determinant of 𝐴𝐴 is defined by
𝑎𝑎11 𝑎𝑎12
det(𝐴𝐴) = |𝐴𝐴| = �𝑎𝑎 𝑎𝑎22 � = 𝑎𝑎11 𝑎𝑎22 − 𝑎𝑎12 𝑎𝑎21
21
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