Matrices Determinants Slides

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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Theoretical Models in Computing

Dr. Mai Duc Thanh1

1Department of Mathematics, International University of Hochiminh City

Lecture 2: Matrices and Determinants

Dr. Mai Duc Thanh Theoretical Models in Computing

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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Outline

1 Matrices


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Outline

1 Matrices

2 Determinants


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Outline

1 Matrices

2 Determinants

3 The inverse matrix


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Example

An animal breeder can buy three types of food for pigs. Eachcase of Brand A contains 25 units of fiber, 30 units of protein,and 30 units of fat. Each case of Brand B contains 50 units of fiber, 30 units of protein, and 20 units of fat. Each case of Brand C contains 75 units of fiber, 30 units of protein, and 20units of fat.How many cases of each should the breeder mix together to

obtain a food that provides 1,200 units of fiber, 600 units of protein, and 400 units of fat?


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Example


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Example

We have the following table

Brand A Brand B Brand C

Fiber 25 50 75Protein 30 35 40Fat 30 25 20

x cases y cases z cases

Let x , y , z represent number of cases of Brands A, B, C,

respectivelyTotal amount of fiber is to be 1,200 units:

25x + 50y + 75z = 1, 200. (1)


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Example

Total amount of protein is 600:

30x + 35y + 40z = 600. (2)

Total amount of fat is 400:

30x + 25y + 20z = 400. (3)

Problem: find x , y , z satisfying the system

25x + 50y + 75z = 1, 20030x + 35y + 40z = 600

30x + 25y + 20z = 400.

(4)


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Example

⇒ We need some data in the form

A =

25 50 75 1, 200

30 35 40 60030 25 20 400

.


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Definition

Many practical applications of engineering and science,quantitative problems of business and economics, andmathematical models involve data of the form

A =

a11 a12 · · · a1n

a21 a22 · · · a2n

· · · · · · · · · · · ·am1 am2 · · · amn

(5)

This array of numbers enclosed by brackets is called an m × nmatrix with m rows and n columns. The entry aij denotes theelement in the ith row and jth column.


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Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Square Matrices, Row and Column Vectors

If m = n then A is called a square matrix of order n

b =

b 1b 2...

b n

If A has only one column (n = 1) then A is called a columnvector

If A has only one row (m = 1) then A is called a row vector

c = (c 1, c 2, ..., c n)


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrices: Principle Diagonal

If A is a square matrix of order n, the diagonal containingelements a11, a22, ..., ann is called the principle, main orleading diagonal.

trace A = a11 + a22 + ...ann =n

i =1

aii

A diagonal matrix is a square matrix that has its onlynon-zero elements along the leading diagonal:

a11

0 0 · · · 00 a22 0 · · · 00 0 a33 · · · 0

· · · · · · · · · · · ·0 0 0 · · · ann


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Identity Matrix, Null Matrix

A special case of diagonal matrices: Unit matrix or identitymatrix I = I n for which a11 = a22 = ... = ann = 1:

I n =

1 0 0 · · · 00 1 0 · · · 00 0 1 · · · 0

· · · · · · · · · · · ·0 0 0 · · · 1

Zero or null matrix 0 is the matrix with every element zero


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Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Transpose Matrix

Transposed matrix AT of A by (5) is the matrix withelements aT

ij = a ji :

AT =

a11 a21 · · · an1

a12 a22 · · · an2· · · · · · · · · · · ·a1n a2n · · · amn

If a square matrix such that AT = A, then aij = a ji . So

elements are symmetric about main diagonal.If AT = A, then A is called a symmetric matrixIf AT = −A, then A is called a skew-symmetric orantisymmetric matrix


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Basic Operations of Matrices

(a) Equality: Two matrices A and B are equal if all their

elements are the same

A = B ⇔ aij = b ij , ∀1 ≤ i ≤ m, 1 ≤ j ≤ n.


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Matrices

Determinants

The inversematrix


(b) Addition: Addition of matrices is straightforwardIf A = (aij ), B = (b ij ), 1 ≤ i ≤ m, 1 ≤ j ≤ n, then

a11 a12 · · · a1n

a21 a22 · · · a2n· · · · · · · · · · · ·

am1 am2 · · · amn

+

b 11 b 12 · · · b 1n

b 21 b 22 · · · b 2n· · · · · · · · · · · ·

b m1 b m2 · · · b mn

A + B =

a11 + b 11 a12 + b 12 · · · a1n + b 1n

a21 + b 21 a22 + b 22 · · · a2n + b 2n· · · · · · · · · · · ·

am1 + b m1 am2 + b m2 · · · amn + b mn


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Matrices

Determinants

The inversematrix


(c) Multiplication by a scalar: Matrix λA has elements λaij :

A =

a11 a12 · · · a1n

a21 a22 · · · a2n

· · · · · · · · · · · ·am1 am2 · · · amn

then

λA =

λa11 λa12 · · · λa1n

λa21 λa22 · · · λa2n

· · · · · · · · · · · ·λam1 λam2 · · · λamn

(6)


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Dr. Mai DucThanh

Matrices

Determinants

The inversematrix


(d) Properties of transform:

(A + B )T

= AT

+ B T

, (AT

)T

= A

(e) Basic rules of addition:A + B = B + A

(A + B ) + C = A + (B + C )λ

(A + B ) =λ

A +λ

B


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Matrix Multiplication

If A is an m × p matrix with elements aij and B is a p × n

matrix with elements b ij , then we can define the product

C = AB as the m × n matrix with entries

c ij =

p k =1

aik b kj i = 1, 2, ..., m, j = 1, 2, ..., n


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Theoretical

Models inComputing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Example:

Given

A =

1 2 −10 3 2

, B =

−1 33 10 2

,

Find AB , BA.


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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Properties of matrix multiplication

A(BC ) = (AB )C

(mA)B = A(mB ) = mAB (A + B )C = AC + BC , A(B + C ) = AB + AC

I mA = AI n = A

(AB )T = B T AT


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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Definition of Determinants

Given square matrices

A =

a11 a12

a21 a22

, B =

a11 a12 a13

a21 a22 a23

a31 a32 a33

Determinant of A, denoted by det A or |A|, is

|A| =

a11 a12

a21 a22

= a11a22 − a12a21

Determinant of 3 × 3 matrix B is

|B | = a11

a22 a23

a32 a33

− a12

a21 a23

a31 a33

+ a13

a21 a22

a31 a32


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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Definition of Determinants

Special case: if A = (a), then det A = a

Example: Evaluate the third-order determinant

1 2 −10 3 21 −1 0


C

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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Determinants: Cofactors

If we take a determinant and delete row i and column j , thedeterminant remaining is called the minor M ij .In general, we take any row (or column) and evaluate an n × n

determinant |A| as

|A| =n

j =1

(−1)i + j aij M ij

Aij = (−1)i + j M ij is called the cofactor of element aij Thus,

|A| =

n j =1

aij Aij


P i f d i

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Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Properties of determinants

(a) If two rows (or two columns) of a determinant are equal,then determinant is zero(b) Multiply a row (or a column) by a scalar:

|B | =λa11 λa12 λa13

a21 a22 a23

a31 a32 a33

= λ|A|

(c) Interchanging two rows (or two columns) changes the signof determinant

(d) Adding multiples of rows (or columns) together makes nodifference to the determinant(e) Transpose: |AT | = |A|(f) Product: |AB | = |A||B |


Adj i M i

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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Adjoint Matrices

The adjoint matrix of a matrix A is the transpose of the matrix

whose entries are cofactors of A:

adj A =

A11 A21 A31

A12 A22 A32

A13 A23 A33

Theorem: A(adjA) = |A|I .Thus,

|A||adj A| = |A(adj A)| = ||A|I n| = |A|n

If |A| = 0, this yields Cauchy theorem:

|adjA| = |A|n−

1

Alsoadj(AB ) = (adjB )(adj)A

If |A| = 0, then A is called singular. If |A| = 0, then A is called

nonsingular Dr. Mai Duc Thanh Theoretical Models in Computing

I f t i

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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Inverse of a matrix

Given a square matrix A, if we can construct a matrix B such

thatAB = BA = I

then we say: B is the inverse of A and write B = A−1

Since A(adjA) = |A|I , if |A| = 0 then

A−1 = 1|A|

adjA

If |A| = 0, then A−1 does not existIf the inverse A−1 exists, then it is unique.Indeed, if AC = CA = I , then

C = CI = C (AA−1) = (CA)A−1 = IA−1 = A−1

Inverse of a product:

(AB )−1 = B −1A−1


E l

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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Example:

Let

1 2 2

2 1 22 2 1

show that A2 − 4A − 5I = 0 and hence A−1 = 15 (A − 4I ).

Calculate A−1 from this result.


H k A i t N2

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Computing

Dr. Mai DucThanh

Matrices

Determinants

The inversematrix

Homework Assignment N2

Textbook: Glyn James, Modern Engineering Mathematics,Addition-Wesley, 2001.

-Pages 279-280: 5, 6, 8, 11, 14, 17-Pages 294: 20, 21, 22, 23, 26, 27, 30, 31-Pages 298-299: 34, 35, 36, 37Deadline: 31st March, 2009


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