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 & c d 0 d c 0 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 row and column. Its sign is (−1)k, where k is 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)

" # The determinant of a b 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