Chapter 2 = Determinants

MATH 264 Linear Algebra

DeterminantsIn this chapter we will study “determinants” or, more

precisely, “determinant functions.” Unlike real-valued functions, such as f(x)=x2, that assign a real number to a real variable x, determinant functions assign a real number f(A) to a matrix variable A.

Although determinants first arose in the context of solving systems of linear equations, they are rarely used for that purpose in real-world applications.

Theorem:det(A) is a function from the set of 2 x 2 matrices to the real numbers having the properties: If you exchange two rows the determinant changes sign ↑ The determinant is linear in each row ↓

𝒅𝒆𝒕 [𝒂 𝒃𝒄 𝒅 ]=−𝒅𝒆𝒕 [𝒄 𝒅

𝒂 𝒃 ]

Another Theorem:For any positive integer n there is exactly one function det(A) from the set of all n x n matrices to the real numbers called the determinant of A having 3 properties:1) The determinant of the identity matrix is 12) If you exchange 2 rows of A the determinant chages

sign.3) The determinant is linear in each row.

Theorem: Determinants of Elementary Matrices

(*)

Minor Determinants is computing the determinant as a linear combination of the determinants of smaller sub-matrices.

In (*) we deleted the first row of A in each of the 3 sub-matrices and then used the entries from the first column as the coefficients of minor determinants. The resulting formula for the determinant of A is called its Cofactor Expansion along the first row.

Examples:1) The identity matrix is a diagonal matrix.2) A square matrix in REF is upper triangle.3) Elementary matrices that scale a row are diagonal

matrices.4) Elementary matrices that add a multiple of an upper

row to a lower row are lower trianglar. Elementary matrices that add a multiple of an lower row to a upper row are lower trianglar.

5) Elementary matrices that exchange 2 rows are neither upper or lower trianglar,

Theorem:1) The transpose of a lower triangular matrix is upper

triangular and the transpose of an upper triangular matrix is lower triangular.

2) A product of lower triangular matrices is lower triangular and the product of upper triangular matrices is upper triangular.

3) A triangular matrix is invertible if and only if all of its diagonal entries are non-zero.

4) The inverse of an invertible lower triangular matrix is lower triangular and the inverse of an invertible upper triangular matrix is upper triangular.

Proof of Theorem:

LU Decomposition:

Example:

Step repeated

Theorem:If A is an invertible matrix that can be reduced to REF without row exchanges then there exists an invertible lower triangular matrix L and invertible upper triangular matrix U with 1s along the diagonal such that

A = LUThe factorization is called an LU-Decomposition of A.

Solving Systems of Linear Equationsusing LU-Decomposition method:

Questions to Get DoneSuggested practice problems (11th edition) Section 2.1 #15-21 oddSection 2.2 #5-21 oddSection 2.3 #7-17 odd