Chapter Four: Determinants

1. Chapter Four: Determinants • 4.I. Definition • 4.II. Geometry of Determinants • 4.III. Other Formulas • Topics: • Cramer’s Rule • Speed of Calculating Determinants • Projective Geometry Ref: T.M.Apostol, “Linear Algebra”, Chap 5.

2. If a (square) matrix T is non-singular, then • Tx= b has a unique solution. • T is row equivalent to I. • Rows of T are L.I. • Columns of T are L.I. • Any map T represents is an isomorphism. • The inverse T 1 exists. Goal: devise det T s.t. T is non-singular  det T 0. Approach: Define determinant as a multi-linear, antisymmetric function of the rows that maps a square matrix to a number.

3. 4.I. Definition 4.I.1. Exploration 4.I.2. Properties of Determinants 4.I.3. The Permutation Expansion 4.I.4. Determinants Exist

4. 4.I.1. Exploration Skipped.

5. 4.I.2. Properties of Determinants Definition 2.1: Determinant function det: Mnn → R s.t. Note: 2. is redundant since

6. Lemma 2.3: • A matrix with two identical rows has a determinant of zero. • A matrix with a zero row has a determinant of zero. • A matrix is nonsingular  its determinant is nonzero. • The determinant of an echelon form matrix is the product down its diagonal. Proof: Straightforward, main argument being that Gaussian reduction without row interchanges & scalar multiplications leaves the determinant unchanged. Example 2.4a:

7. Example 2.4b:

9. Example 2.4: Example 2.5: Lemma 2.6 : For each n, if there is an nn determinant function then it is unique. Proof: Reduced echelon form is unique.

10. Exercises 4.I.2. 1. Show that 2. Refer to the definition of elementary matrices in the Mechanics of Matrix Multiplication subsection. (a) What is the determinant of each kind of elementary matrix? (b) Prove that if Eis any elementary matrix then |ES| = |E| | S| for any appropriately sized S. (c) (This question doesn’t involve determinants.)Prove that if Tis singular then a product TSis also singular. (d) Show that |TS| = |T| | S| . (e) Show that if Tis nonsingular then |T1| = |T| 1 .

11. 4.I.3. The Permutation Expansion Definition 3.2: Multilinear Maps Let V be a vector space. A map f : V n→R is multi-linearif Lemma 3.3: Determinants are multilinear. Proof: The definition of determinants gives (2) so we need only check (1), i.e., that where v + w, v, w are the ith rows of the respective determinants. Let be a basis of the row space

12. Since adding aρk to ρi doesn’t change the value of a determinant, we have

13. Example 3.4:

14. Example 3.5:

15. Definition 3.7: n-permutation An n-permutationis a sequence consisting of an arrangement of the numbers 1, 2, . . . , n. Definition 3.9: Permutation Expansionfor Determinants The permutation expansionfor determinants is where P(1) P(2) … P(n) denotes the permutation P of { 1, 2, …, n } and ()P equals to +1 (1) if P is an even (odd) permutation.

16. Example 3.10: Theorem 3.11: For each n there is a nn determinant function. Proof: Deferred to next section. Proof: Deferred to next section. Theorem 3.12: det A = det AT Corollary 3.13: A matrix with two equal columns is singular. Column swaps change the sign of a determinant. Determinants are multilinear in their columns. Proof: Columns of A are just rows of AT.

17. Exercises 4.I.3. 1. Show that if an nn matrix has a nonzero determinant then any column vector vRncan be expressed as a linear combination of the columns of the matrix. 2. Show that if a matrix can be partitioned as where J, K are square, then | T | = | J | | K |. 3. Show that

18. 4.I.4. Determinants Exist Skipped