Chapter 2 Determinant Objective: To introduce the notion of determinant, and some of its properties as well as applications.

Test for singularity of a matrix instead of by definition. Find the area of a parallelogram generated by two vectors. Find the volume of a parallelopipe spanned by three vectors. Solve Ax=b by Cramer’s rule. Several applications of determinant

Introduction to Determinant (to determine the singularity of a matrix) Consider. If we define det(A)=a, then A is nonsingular.

 Let Suppose, then A If we define then A is nonsingular Case2 2×2 Matrices

Suppose but, then A and Thus A is nonsingular Suppose A is singular & det(A)=0. To summarize, A is nonsingular Case2 2×2 Matrices (cont.)

 Let -Suppose, A Case3 3×3 Matrices

 From 2x2 case, A I Then A is nonsingular Case3 3×3 Matrices (cont.) define

that A I Easily Shown for Cases

 For, where Recall

 For, where,, Recall (cont.)

Definition: Let,, and let the matrix obtained from A by deleting the row & column containing The is called the minor of The cofactor of is denoted as Generalization

Definition: The determinant of is defined as, if n=1, if n>1 Note: det is a function from to.

Theroem2.1.1:Let, Hint: By induction or sign-type definition.

Theroem2.1.2: Let,and Pf: By induction, n=1,ok! Suppose the theorem is true for n=k. If n=k+1, By induction The result then follows.

Theroem2.1.3: Let be a triangular matrix. Then Hint:expansion for lst row or column and induction on n. Theroem2.1.4: (i)If A has a row or column consisting entirely of zeros, then (ii)If A has two identical rows or columns, then Hint for (ii): By mathematical induction.

Note that For example,, Question: Is §2-2 Properties of Determinants

Lemma2.2.1: Let, then

Pf: Case for i=j follows directly from the definition of determinant. For, define to be the matrix obtained from A by replacing the j th row of A by i th row of A. (Then has two identical rows) expansion along jth row Proof of Lemma2.2.1

Proof of Lemma2.2.1 (cont.) j th row

Note that  by Th  先對非交換列展開 數學歸納法

    Lemma 2.2.1



Thus, we have If E is an elementary matrix In fact, det(AE)=det(A)det(E) Question:

Theorem2.2.2: is singular Pf:Transform A to its row echelor from as If A is singular If A is nonsingular The result then follows.

Theorem2.2.3:Let.Then Pf: If B is singular AB is singular If B is nonsingular

 Objective: Use determinant to compute and solve Ax=b. §2-3 Cramer ’ s Rule

Def: Let.The adjoint of A is defined to be where are cofactor of The Adjoint of a Matrix

By Lemma2.2.1, we have If A is nonsingular, det(A) is a nonzero scalar

For a 2×2 matrix : If A is nonsingular, then Example 1 (P.116)

Q: Let, compute adj A and A -1. Sol: Example 2 (P.116)

Theorem2.3.1:(Cramer’s Rule) Let be nonsingular and. Denote the matrix obtained by replacing the ith column of A by.Then the unique sol. of is Pf:

Q: Use Cramer’s rule to Solve Example 3 (P.117)

Sol: Example 3 (cont.)

 Let.Then volume of the parallelopipe spanned by and is  Let.Then the area of the parallelogram spanned by and is

For example, the message Send Money might be coded as 5, 8, 10, 21, 7, 2, 10, 8, 3 here the S is represented by a “5”, the E is represented by a “8”, and so on. Application 1: Coded Message (P.118)

Application 1: Coded Message (cont.) If A is a matrix whose entries are all integers and whose determinants is ± 1, then, since, the entries of A -1 will be integers. Let

Application 1: Coded Message (cont.) We can decode it by multiplying by A -1 We can construct A by applying a sequence of row operations on identity matrix. Note: A -1 AB(encoding Message)