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Unit 5

- Determinants and Matrices

5.1 Definition of Determinants

- (A) A square array of real (or complex) numbers

arranged in n rows and n columns is called a

determinant of the nth order .

5.1 Definition of Determinants

- The number in the determinant is called an

element or entry of the determinant. There are n2

elements in a determinant of the nth order.

Usually, we use small letters to denote elements.

The symbol aij denotes the element located in

the ith row, the jth column.

e.g. a34 located in 3th row

4th column

5.1 Definition of Determinants

- (B) The value of a determinant of order 2 is

defined by

The expression a11a22 - a12a21 on the right hand

side of (1) is called the expansion of the second

order determinant.

5.1 Definition of Determinants

- (C) The value of a determinant of order 3 is

defined by

The expression on the right hand side of (2) is

called the expansion of the third order

determinant.

5.2 Properties of Determinants

- (1) The value of a determinant remains unchanged

if all the rows and columns are correspondingly

interchanged.

5.2 Properties of Determinants

- (2) The determinant changes sign only but its

absolute value remains unchanged if any two rows

(or any two columns) are interchanged.

5.2 Properties of Determinants

- (3) If the elements of a row (resp. a column) are

proportional to those of another row (resp.

another column), then the value of the

determinant is zero.

5.2 Properties of Determinants

- (4) If the elements of a row (resp. a column) are

identical to those of another row (resp. another

column), then the value of the determinant is

zero.

5.2 Properties of Determinants

- (5) If the elements of any row (resp. a column)

of a determinant are multiplied by the same

factor, the resulting determinant is equal to the

product of that factor and the original

determinant.

5.2 Properties of Determinants

- (5 ext) If the elements of a row (resp. a column)

are zero, the value of the determinant is zero.

5.2 Properties of Determinants

- (6) If the elements of any row (resp. a column)

are added or subtracted by equimultiples of the

corresponding elements of another row (resp.

another column), the value of the determinant is

altered.

5.2 Properties of Determinants

- (6) If the elements of any row (resp. a column)

are added or subtracted by equimultiples of the

corresponding elements of another row (resp.

another column), the value of the determinant is

altered.

5.2 Properties of Determinants

- (7) If the elements of a row (resp. a column) of

a determinant consists of an algebraic sum of the

terms, the determinant is equal to the sum of two

other determinants in each of which the elements

consist of single term.

5.2 Properties of Determinants

- (7) If the elements of a row (resp. a column) of

a determinant consists of an algebraic sum of the

terms, the determinant is equal to the sum of two

other determinants in each of which the elements

consist of single term.

P.133 Ex.5A

5.3 Minors and Cofactors

- Let aij be an element of the determinant A

located in the ith row, the jth column. The minor

of aij , is denoted by ?ij, is defined as the

determinant formed by deleting the ith row and

jth column of A.

5.3 Minors and Cofactors

- Let A be a determinant, aij be an element of A in

the ith row, the jth column and ?ij be the minor

of aij. The cofactor of aij is defined by (-1)ij

?ij and is usually denoted by Aij.

5.3 Minors and Cofactors

5.3 Minors and Cofactors

The sum of the products of the elements of any

row (or any column) and their own cofactors is

equal to the value of the determinant.

5.3 Minors and Cofactors

The sum of the products of the elements of any

row (or any column) and their own cofactors is

equal to the value of the determinant.

P.145 Ex.5B

5.4 Factorization of Determinants

To factorize a given determinant, the following

two methods are usually employed

(1) Apply properties of a determinant to

transform the entries in a row (or a column)

until there is a common factor among the entries.

(2) If the determinant is a polynomial in x and

if the determinant vanishes when x a, then (x

- a) is a factor.

5.4 Factorization of Determinants

5.4 Factorization of Determinants

- Method II
- remains unchanged if a, b, c are respectively

replaced by b, c, a.Thus it is a cyclic symmetric

about a, b, c and is homogeneous. - If we put a b or b c or c a, ? 0.
- Hence ? has factors a b, b c and c a.
- As ? is of degree 3, we let ? ? k(a-b)(b-c)(c-a).
- By comparing coefficients of bc2, we get k 1.
- ? ? ? (a-b)(b-c)(c-a)

P.149 Ex.5C

5.5 Definition and Basic Operation of Matrices

Matrix A rectangular array of real (or complex)

numbers arranged in m rows and n column is called

a m x n matrix. A m x n matrix is usually

represented in the form

5.5 Definition and Basic Operation of Matrices

The number aij in the ith row and the jth column

of a matrix is called an element or entry. Hence

a m x n matrix contains mn elements. We call

m x n the order of a matrix, and we usually use

capital letters to denote matrices.

Matrix is a mathematical tool, it is not a number.

5.5 Definition and Basic Operation of Matrices

Equality of matrices Two matrices A (aij) and

B (bij) are equal if and only if they are of

the same orders and aij bij, for all i 1, 2 ,

.., m j 1, 2,..., n.

5.5 Definition and Basic Operation of Matrices

Sum of Matrices The sum of two m x n matrices A

(aij) and B (bij) is the m x n matrix C

(cij), where cij aij bij, for all i 1, 2,

., m j 1, 2, , n.

5.5 Definition and Basic Operation of Matrices

Scalar Multiplication The scalar multiplication

of a m x n matrix A (aij) by a scalar ? (here ?

is a real or a complex number) is the m x n

matrix C (cij), where

cij ? aij, for every i 1, 2, , m j 1,

2,, n. We usually write C ?A. This defines

scalar multiplication of a matrix A by a scalar ?.

5.5 Definition and Basic Operation of Matrices

Negative Matrix Let A be any matrix, the symbol

A denotes (-1)A.

5.5 Definition and Basic Operation of Matrices

Difference of Matrices Let A, B be two matrices

of the same order. The difference A - B is

defined by A - B A (-1)B.

5.5 Definition and Basic Operation of Matrices

For any three matrices A, B, C of the same order

and any scalars ?, ?, we have

- Addition is associative
- (A B) C A (B C)

(2) Addition is commutative A B B A

(3) Scalar multiplication is distributive over

addition and subtraction.

5.5 Definition and Basic Operation of Matrices

Multiplication of Matrices Let A (aik) be a m x

n matrix and B (bkj) be a n x p matrix, the

product AB is the m x p matrix C (cij), where

for all i 1, 2,., m j 1, 2, , p.

The product AB defines multiplication of matrices.

5.5 Definition and Basic Operation of Matrices

Multiplication of Matrices

5.5 Definition and Basic Operation of Matrices

Multiplication of Matrices

5.5 Definition and Basic Operation of Matrices

Multiplication of Matrices

5.5 Definition and Basic Operation of Matrices

Multiplication of Matrices

Multiplication of matrices is non-commutative.

5.5 Definition and Basic Operation of Matrices

Multiplicative Properties of Matrices

- Let A, B, C be three matrices of order m x n, n

x p, p x q respectively, then

(AB)C A(BC)

(2) Let A, B be two matrices of order m x n and

C, D be matrices of order n x p, q x m

respectively, then

(3) Let A, B be two matrices of order m x n and

n x p respectively. For any scalar ?, we have

?(AB) (?A)B A(?B).

5.6 Special Types of Matrices

(1) The square Matrix A n x n matrix is called

a square matrix of order n.

(2) The identity Matrix A square matrix A

(aij) of order n is called an identity matrix or

a unit matrix of order n if and only if

and is denoted by In.

5.6 Special Types of Matrices

(3) The scalar Matrix A square matrix (bij) is

called a scalar matrix if and only if

5.6 Special Types of Matrices

5.6 Special Types of Matrices

(5) The Zero/ Null of Matrix A matrix (not

necessarily square) is called a zero matrix or a

null matrix if and only if all its elements are

zero. A zero matrix of order m x n is usually

denoted by 0mxn.

(i) For any matrix A and a zero matrix, both are

of the same order, A 0 0 A A

(ii) For any square matrix A and a square zero

matrix 0, both are of the same order, A x 0

0 x A 0

5.6 Special Types of Matrices

5.6 Special Types of Matrices

(7) The Idempotent Matrix A square matrix is

called idempotent if and only if A2 A.

5.6 Special Types of Matrices

(8) The transpose of Matrix The transpose of a

m x n matrix A (aij), denoted by AT or At or A,

is the n x m matrix AT (bij), where bij aji.

5.6 Special Types of Matrices

(8 ext.) Properties of the transpose of matrix

5.6 Special Types of Matrices

(9) The Symmetric Matrix A square matrix A is

said to be symmetric if and only if AT A.

5.6 Special Types of Matrices

(10) The skew-symmetric Matrix A square matrix

a is said to be skew-symmetric if and only if AT

- A

5.6 Special Types of Matrices

(11) The Row Vector and the Column Vector A

1 x n matrix is called a row vector.

A m x 1 matrix is called column vector.

P.156 Ex.5D

5.7 Multiplicative Inverse of a Square Matrix

The determinant of A, denoted by det A or A, is

defined as the determinant

5.7 Multiplicative Inverse of a Square Matrix

General properties between determinant and matrix

- For any square matrix A of order n and any scalar

?, det (?A) ?n det A

or

Why is there power n?

5.7 Multiplicative Inverse of a Square Matrix

Verification

5.7 Multiplicative Inverse of a Square Matrix

General properties between determinant and matrix

(2) For any two square matrices A and B of the

same order, det (AB) det A x det B

or

How can we verify this property?

5.7 Multiplicative Inverse of a Square Matrix

Verification

5.7 Multiplicative Inverse of a Square Matrix

Verification

By direct expansion of the determinants, the

identity holds.

5.7 Multiplicative Inverse of a Square Matrix

General properties between determinant and matrix

(3) For any square matrix A, det (AT) det A

or

How can we prove this property?

5.7 Multiplicative Inverse of a Square Matrix

5.7 Multiplicative Inverse of a Square Matrix

The Cofactor Matrix Let A (aij)nxn be a

square matrix. The cofactor matrix of A, denoted

by cof A, is defined by cof A (Aij)nxn, where

Aij is the cofactor of aij, for every i, j 1,

2,,n.

5.7 Multiplicative Inverse of a Square Matrix

The Adjoint Matrix Let A be a square matrix.

The transpose of the cofactor matrix of A, i.e.

(cof A)T, is called the adjoint matrix of A,

denoted by adj A.

5.7 Multiplicative Inverse of a Square Matrix

The Non-singular/Invertible Matrix A square

matrix A of order n is said to be non-singular or

invertible if and only if there exists a square

matrix B such that AB

BA In, where In is an identity matrix of order

n, and the matrix B is called the multiplicative

inverse or simply the inverse of A, which is

denoted by A-1, i.e. AA-1 A-1A In.

The inverse of a non-singular matrix is unique.

5.7 Multiplicative Inverse of a Square Matrix

Lemma For any square matrix of order n,

A(adj A) (adj A)A (det A)In, where In is an

identity matrix of order n.

?

Why does (adjA)A equal (detA)In?

5.7 Multiplicative Inverse of a Square Matrix

(adj A)A (det A)In

Verification

Why do the elements arrange in this way?

5.7 Multiplicative Inverse of a Square Matrix

How can we simplify these three elements?

Then how about the other elements?

Why does each row equal zero?

And

5.7 Multiplicative Inverse of a Square Matrix

5.7 Multiplicative Inverse of a Square Matrix

(adj A)A (det A)In

Verification

5.7 Multiplicative Inverse of a Square Matrix

Lemma For any square matrix of order n,

A(adj A) (adj A)A (det A)In, where In is an

identity matrix of order n.

(adj A)A (det A)AA-1

5.7 Multiplicative Inverse of a Square Matrix

5.7 Multiplicative Inverse of a Square Matrix

On the other hand,

A square matrix A is said to be singular or not

invertible if and only if the inverse, A-1, of A

does not exist.

A square matrix A is singular if and only if det

A 0.

5.8 Properties of Inverses

If A and B are non-singular matrices of the same

order, and any scalar ? we have

(1) If AC1 AC2 or C1A C2A, then C1 C2

(2) A-1 is non-singular and (A-1)-1 A

(3) AB is non-singular and (AB)-1 B-1A-1

Quite similar to (AB)T BTAT We will show the

identity (AB)-1 B-1A-1 later.

(4) For any positive integer n, An is

non-singular, and (An)-1 (A-1)n.

5.8 Properties of Inverses

If A and B are non-singular matrices of the same

order, and any scalar ? we have

(6) AT is non-singular and (AT)-1 (A-1)T.

(7) If AC 0, then C 0.

5.8 Properties of Inverses

If A and B are non-singular matrices of the same

order, and any scalar ? we have

Prove (AB)-1 B-1A-1

5E

P.164 Ex.5E

5.9 Some Illustrative Examples

5.9 Some Illustrative Examples

P.176 Ex.5F

Transformations of Points on the Coordinate Plane

5.10 Linear Transformations on the Rectangular

Cartesian Plane

5.10 Linear Transformations on the Rectangular

Cartesian Plane

5.10 Linear Transformations on the Rectangular

Cartesian Plane

5.10 Linear Transformations on the Rectangular

Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

(IV) Reflection

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

(IV) Reflection

The matrix representing the reflection about

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

5.11 Some special Linear Transformations on the

Rectangular Cartesian Plane

P.195 Ex.5G