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

If a (square) matrix T is non-singular, then T x = 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.

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.I.1. Exploration Skipped.

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

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:

Example 2.4b:

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.

Exercises 4.I 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 E is any elementary matrix then |ES| = |E| | S| for any appropriately sized S. (c) (This question doesn’t involve determinants.) Prove that if T is singular then a product TS is also singular. (d) Show that |TS| = |T| | S|. (e) Show that if T is nonsingular then |T  1 | = |T|  1.

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-linear if 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. Letbe a basis of the row space

Since adding aρ k to ρ i doesn’t change the value of a determinant, we have

Example 3.4:

Example 3.5:

Definition 3.7: n-permutation An n-permutation is a sequence consisting of an arrangement of the numbers 1, 2,..., n. Definition 3.9: Permutation Expansion for Determinants The permutation expansion for 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.

Example 3.10: Theorem 3.11: For each n there is a n  n determinant function. Proof: Deferred to next section. Theorem 3.12:det A = det A T Proof: Deferred to next section. 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 A T.

Exercises 4.I Show that if an n  n matrix has a nonzero determinant then any column vector v  R n can 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

4.I.4. Determinants Exist Skipped