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Lecture May 2011

Matrices and Determinants

Mathematics XI 2011

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1.1 Matrices1.2 Operations of matrices1.3 Types of matrices1.4 Properties of matrices1.5 Determinants

1.61.7

Inverse of a 33 matrixApplication

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Zero matrices

Every element of a matrix is zero, it is called a zero matrix, i.e.,

0 0 0

0 0 0

0 0 0

A

1.1 Matrices

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Sums of matrices

1.2 Operations of matrices

Two matrices of the same order are said to be conformable for addition or subtraction.

Two matrices of different orders cannot be added or subtracted, e.g.,

are NOT conformable for addition or subtraction.

2 3 7

1 1 5

1 3 1

2 1 4

4 7 6

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1.3 Types of matrices

Identity matrix

The inverse of a matrix

The transpose of a matrix

Symmetric matrix

Orthogonal matrix

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(AB)-1 = B-1A-1

(AT)T = A and (A)T = AT

(A + B)T = AT + BT

(AB)T = BT AT

1.4 Properties of matrix

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1.5 Determinants

Consider a 2 2 matrix: 11 12

21 22

a aA

a a

Determinant of order 2

Determinant of A, denoted , is a number and can be evaluated by

11 1211 22 12 21

21 22

| |a a

A a a a aa a

| |A

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1.6 Inverse of a 33 matrix

Cofactor matrix of 1 2 3

0 4 5

1 0 6

A

The cofactor for each element of matrix A:

11

4 524

0 6A 12

0 55

1 6A 13

0 44

1 0A

21

2 312

0 6A 22

1 33

1 6A 23

1 22

1 0A

31

2 32

4 5A 32

1 35

0 5A 33

1 24

0 4A

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Cofactor matrix of is then given by:

1 2 3

0 4 5

1 0 6

A

24 5 4

12 3 2

2 5 4

1.6 Inverse of a 33 matrix

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1.6 Inverse of a 33 matrix

Inverse matrix of is given by:1 2 3

0 4 5

1 0 6

A

1

24 5 4 24 12 21 1

12 3 2 5 3 522

2 5 4 4 2 4

T

AA

12 11 6 11 1 11

5 22 3 22 5 22

2 11 1 11 2 11

1.7 Applications Cement Steel Floor(sq.ft) Banglow 200 Yds 300 2500 2000 Banglow 300 Yds 500 3500 3000 Banglow 500 Yds 750 5000 5500

Khi Lhr Isl 450 375 520330 350 410 120 90 150

Total Cost

Khi Lhr Isl

Banglow 200 Yds 1200000 1167500 1481000 Banglow 300 Yds 1740000 1682500 2145000 Banglow 500 Yds 2647500 2526250 3265000

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Students can also create parallel examples in

Financial institutionIndustriesTime management

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By : Syed Shujaat Hussain

Special Thanks to My Master Trainer Mr.

Adnan Kiyani

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