Determinants: Definitions The Key Fact Fact. A square matrix is - PowerPoint PPT Presentation

Determinants: Definitions

The Key Fact Fact. A square matrix is invertible if and only if its determinant is not zero. detONE: 2

The 2 × 2 Case � � � a b � a b � � det = � = ad − bc � � c d c d � We use both det and vertical lines to indicate determinant. detONE: 3

Permutation Matrices Defn. A permutation matrix is a square ma- trix that contains only 0 ’s and 1 ’s with exactly one 1 in each row and column. Defn. A generalized permutation matrix is square matrix with at most one nonzero element in each row and column. detONE: 4

The Sign of a Permutation Matrix Defn. The sign of a generalized permutation matrix is ( − 1) k , where k is the number of row interchanges needed to change the matrix to be diagonal. detONE: 5

Definition of Determinant Defn. The determinant of matrix is defined by: construct all possible generalized permutation matrices it contains and for each, multiply the relevant entries together then by the sign, and then sum the results. For example: � a b � � a 0 � � 0 b � has gen-perm matrices & 0 d c 0 c d The former has positive sign; the latter has neg- ative sign. So we get ad − bc . detONE: 6

Some Consequences Fact. If matrix A has an all-zero row or col- umn, then det A = 0 . Fact. The determinant of a triangular matrix is the product of the diagonal entries. In particular, the determinant of the identity matrix I is 1 . detONE: 7

Summary A generalized permutation matrix is square ma- trix with at most one nonzero element in each Its sign is ( − 1) k , where k is row and column. number of row interchanges needed to change it to diagonal. The determinant of a matrix is defined by: con- struct all possible generalized permutation ma- trices it contains and for each, multiply the rele- vant entries together then by the sign, and then sum the results. detONE: 8

Summary (cont) � � a b The determinant of is ad − bc . The determi- c d nant of a triangular matrix is the product of the diagonal entries. A square matrix is invertible if and only if its determinant is not zero. detONE: 9