🗊Презентация Determinants

Determinants

1 The Determinant of a Matrix 2 Properties of Determinants 3 Application of Determinants: Cramer’s Rule

Given a square matrix A its determinant is a real number associated with the matrix. Given a square matrix A its determinant is a real number associated with the matrix. The determinant of A is written: det (A) or |A| For a 2x2 matrix, the definition is

※ The determinant is NOT a matrix operation ※ The determinant is NOT a matrix operation ※ The determinant is a kind of information extracted from a square matrix to reflect some characteristics of that square matrix ※ For example, this chapter will discuss that matrices with a zero determinant are with very different characteristics from those with non-zero determinants ※ The motives to calculate determinants are to identify the characteristics of matrices and thus facilitate the comparison between matrices since it is impossible to investigate or compare matrices entry by entry ※ The similar idea is to compare groups of numbers through the calculation of averages and standard deviations ※ Not only the determinant but also the eigenvalues and eigenvectors are the information that can be used to identify the characteristics of square matrices

The determinant of a 2 × 2 matrix: The determinant of a 2 × 2 matrix:

Historically speaking, the use of determinants arose from the recognition of special patterns that occur in the solutions of linear systems: Historically speaking, the use of determinants arose from the recognition of special patterns that occur in the solutions of linear systems:

Determinants 2x2 examples

For a matrix For a matrix

For a matrix For a matrix

For a matrix For a matrix

For the matrix For the matrix

For the matrix For the matrix

Ex: Ex:

Theorem: Expansion by cofactors Theorem: Expansion by cofactors

Ex: The determinant of a square matrix of order 3 Ex: The determinant of a square matrix of order 3

Ex: The determinant of a square matrix of order 3 Ex: The determinant of a square matrix of order 3

Ex: The determinant of a square matrix of order 4 Ex: The determinant of a square matrix of order 4

Sol: Sol:

Upper triangular matrix: Upper triangular matrix:

Theorem: (Determinant of a Triangular Matrix) Theorem: (Determinant of a Triangular Matrix)

Ex: Find the determinants of the following triangular matrices

2 Properties of Determinants Conditions that yield a zero determinant

Notes: Notes:

Ex 2: Ex 2:

Ex 3: Classifying square matrices as singular or nonsingular Ex 3: Classifying square matrices as singular or nonsingular

Inverse Matrices

Theorem of Inverse Matrices

Example 3

Ex 4: Ex 4:

The similarity between the noninvertible matrix and the real number 0 The similarity between the noninvertible matrix and the real number 0

If A is an n × n matrix, then the following statements are equivalent If A is an n × n matrix, then the following statements are equivalent

Ex 5: Which of the following system has a unique solution? Ex 5: Which of the following system has a unique solution?

Sol: Sol:

3 Applications of Determinants Theorem: Cramer’s Rule

Ex: Use Cramer’s rule to solve the system of linear equation Ex: Use Cramer’s rule to solve the system of linear equation

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