Lecture 27 - Mar 13

Lecture 27 - Mar 13 - Friday March 13 Lecture 27 :...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Friday March 13 Lecture 27 : Determinants (Refers to 5.1) Expectations : 1. Define the ( i , j )- minor of a square matrix. 2. Define the ( i , j )- cofactor of a square matrix. 3. Calculate the determinant of a square matrix by row or column expansion. 4. Apply the fact that the determinant of a triangular matrix is the product of the entries on the diagonal. 5. Recognize that a triangular matrix A is invertible iff its determinant is not 0. 6. Recognize the effects the 3 elementary row (column) operations have on the value of the determinant of a matrix. 7. Recognize that det( A ) = det( A T ). 8. Recognize conditions on A that force det( A ) = 0. 9. Find determinants by using the properties of determinants. 27.1 Definition A determinant is a function D which maps a square matrix A to a real number according to well defined formula applied to the entries of the matrix. We denote this function by det( A ) or | A | . In the case of a 2 × 2 matrix A = [ a ij ] 2 × 2 , det( A ) = a 11 a 22 a 12 a 21 . The general definition for the determinant is of an n × n matrix is given after we have developed some terminology. 27.2 Definition - Given a square n by n matrix A , for the entry a ij in A , we associate a matrix M ij of dimension n 1 by n 1 whose entries are the ones that remain when we remove from A the row and the column containing the entry a ij . We call the number m ij = det( M ij ), the minor of the element a ij , or the ( i, j ) minor . 27.2.1 Remark For an n × n matrix A there are n 2 minors m ij , one for each entry of A . 27.3 Definition - For each minor m ij , we define the cofactor c ij as c ij = ( 1) i + j m ij . 27.3.1 Examples are given.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
27.4 Definition - For an n × n matrix A = [ a ij ] n x n , the determinant of A , det( A ) = | A | is defined as : Det( A ) = j = 1 to n a 1j c 1j = a 11 c 11 + a 12 c 12 + . .. + a 1 n c 1 n . 27.4.1
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 7

Lecture 27 - Mar 13 - Friday March 13 Lecture 27 :...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online