Chapter 2 Determinants MATH 264 Linear Algebra Determinants

Chapter 2 = Determinants MATH 264 Linear Algebra

Determinants In this chapter we will study “determinants” or, more precisely, “determinant functions. ” Unlike real-valued functions, such as f(x)=x 2, 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:

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 determinantof A having 3 properties: 1) The determinant of the identity matrix is 1 2) 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 Determinantsis 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 submatrices 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 Expansionalong 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 1 s along the diagonal such that A = LU The factorization is called an LU-Decomposition of A.