Lecturer fkb 10103 lesson 0 matrices issuu by elizabethsoo71@gmail.com

FKB10103 ENGINEERING TECHNOLOGY MATHEMATICS 1

LINEAR ALGEBRA (MATRICES) 2

Content • Recall Matrices (introduction, terminology, matrix algebra) • Determinant

• Elementary Row Operations • Inverse Matrix 3

By the end of this lesson, students will be able to : * Evaluate the determinant of matrix 2x2, 3x3 & etc.. * Obtain the inverse matrix

* Perform row operations

4 4

Matrices are used in control theory, electrical principles, vibrations, structural analysis, etc.

Many engineering problems can be written in terms of simultaneous equations and if there are a large number of equations then, it is easier to use matrices rather than substitution method 5 or elimination method

EXAMPLES The currents i1 , i2 and i3 in the star-connection shown in the figure are given by the following simultaneous equations :

Z1i1  Z 2i2  e1  e2 Z 2i2  Z 3i3  e2  e3 i1  i2  i3  0 Solve the equations for the three currents by using Cramer’s rule. 6

Another application of matrices is in image processing. Many displays form images by lighting up tiny dots, called pixels, on their screens.

Figure below shows a symbol represented by lighting up the appropriate pixels. If we let 1=‘pixel on’ and 0=‘pixel off’ Then we can represent this in matrix form as : 1  0 0   1

1 rows

1  0 0   1  45 columns

Introduction A matrix is an array of numbers enclosed in a bracket. It is used to store information. If it has m rows and n columns then it is said to be an m by n matrix. Column

 a 11 a12   a 21 a 22 A m n     Dimension of  the matrix A  a  m1 a m2

 a 1n    a 2n        a mn 

Row

Some Special Matrices Square matrix

Zero matrix

 1.2  2.6   A    0.4 1.7  22

 0 0  B    0 0  22

Row matrix C  1 2 313

Column matrix 1 D     2  21 10

Some Special Matrices Diagonal matrix, 1 0  A    0 2  22

1 0 0   B  0 2 0  0 0 3  33

Identity matrix, I n

1 0  I 2   0 1

1 0 0   I3   0 1 0  0 0 1   11

Matrix Algebra Matrices can be combined in various ways: • • • •

Addition Subtraction Multiplication by a Scalar Multiplication of Matrices

Suppose that A and B are two matrices of the same size . The sum C=A+B is defined by: 1   2

3 5    4 6

7  1 5   2  6 8 

3  7 6     4  8  8

10    12 

The difference C=A-B is defined by: 1   3

9  0    9  2

7 1 0     8 3  2

9  7 1     9 8 1 13

2   1

If  is a scalar and A is a matrix then A is the matrix of the same size as A given by λ A ij  λ A ij 1 3  2

3   3 1    4  3 2

3 3  3    6 3 4 

9   12 

14 14

Transpose of a matrix can be obtained by interchange the row with the column

 3 5   3 4      4 2    5 2 T

15 15

2   3

1 4    7 5

9  2 4    8  3 5

1 9    7  8

WRONG 16

Matrix multiplication is row x column : column row

a   c

b e    d g

 ae  bg    ce  dg

f    h af  bh    cf  dh  17

1   2

2 0

4 1   0  1 23  2

2  1 1  32

 1 4  2  0  1 2    2  4  0  0  1 2 4  0  2   8  0  2

1 2  2  1  1 1    2  2  0  1  1  1  6 5 2  2  1       4  0  1 10 5  22

We can only multiply matrices if the number of columns of the 1st matrix = the number of rows of 2nd matrix 18

Properties of Matrix Algebra • Multiplication of matrix (Not Commutative): AB ≠ BA

• Identity Matrix: AI = IA = A • Zero Matrix: A0 = 0A =0 A + 0 =A

• Transpose of Matrix:

(AB)T

BTAT

Determinants If A is square matrix then the determinant function is denoted by det and det(A) is defined to be the sum of all the signed elementary matrices of A. a 11

det A  A 

a 12  a 1n

a 21 a 22  a 2n 







The result of a determinant is a single number.

a n1 a n2  a nn 20

Finding Determinants A

Case 1 : Matrix (2x2)

3 5 4

Case 2 : Matrix (3x3) Method 1 : Diagonal Method (copy the first 2 columns at the back of the matrix)

det A  A 

2 21

Finding Determinants Case 2 : Matrix (3x3) Method 2 : Cofactor Method + +  2 2 0    A  0 2 4   1 1 1   

Step 1 :Choose any row/column with maximum zero

Step 2 : (sign)(element)(minor)

 2M11   2M12  0M13  2 2 4 1

1

2

0 4 1

1

0

Finding Determinants Case 3 : Matrix (4x4), (5x5),(6x6)……….. Method : Cofactor Method + - + -

 2 2 0 3     0 2 4 1  A 1 1 1 2    0 2  3 4  

Step 1 :Choose any row/column with maximum zero Step 2 : (sign)(element)(minor)

A  2M11   2M12  0M13  (3)M14 2

2 4 1 1

1 2

2

4

1

0

0 2 4 3 1

1

3 23

Finding Determinants “Diagonal Method” Determinant function for a 2×2 matrix.

det A  A 

a 11

a 12

a 21 a 22

 a 11a 22  a 21a 12 24

Example 1 Compute the det(A) for the following matrix.

 3 5   A   4 2  det A  

3 5 4

 6   20  26 25

Finding Determinants “Diagonal Method” Determinant function for a 3×3 matrix.

a 11 a 12 B  a 21 a 22 a 31 a 32

a 13 a 23 a 33

 a 11a 22 a 33  a 12 a 23a 31  a 13a 21a 32  a 31a 22 a 13  a 32 a 23a 11  a 33a 21a 12 26

Example 2 Compute the determinate for the following matrix.

2 3 3   A 0 4 1 1 2 1  

 2 2   B B  0 0  1 1  

32 3 0    42 1 4  1 12 1   

Example 2…solution det A  A 

 8  3  0  12  4  0  -5

Example 2…solution B 

2 -2 0 0 -2 -4 1 1 -1

 4   8  0   0   - 8  0 

B  20 29

Minor & Cofactor If A is a square matrix then the minor of a i j , denoted by M i j , is the determinant of the submatrix that results from removing the ith row and jth column of A.

If A is a square matrix then the cofactor of denoted by C i j , is the number

a i j,

Ci j   1 M i j i j

Example 3 For the following matrix compute the minor, M23 and the cofactor, C23

 3 1 6    A   9 5 2  0 4 7  

Example 3…solution Minor

1 6

M 23  9  5 2  0 Cofactor

1

3 1 0

 12

C 2 3   12  3 M 2 3  12 32

Finding Determinants “Method of Cofactors” If A is an n×n matrix.

A 

a 11

a 12  a 1n

a 21 a 22  a 2n 







a n1 a n2  a nn • Choose any row, then:

A  a i1Ci1  a i 2Ci 2  .....  a i n Ci n

• Choose any column, then:

cofactor

A  a1 jC1 j  a 2 jC2 j  .....  a n jCn j

Example 4 Compute the det(A) for the following matrix.

 2 2 0    A  0 2 4   1 1 1   

Example 4…solution 2  2 0   Choose any row/column   A  0 2 4  with maximum zero:   1 1  1 Step-by-step :  

A  2C11   2C12  0C13

 2 M11   2- M12  0 M13   2 2 4 1

1

2

0 4 1

1

0

 26  24  0  20 35

Example 4…solution 2  2 0   Choose any row/column   A  0 2 4  with maximum zero:  1 1 1    Short-cut : (sign)(element)(minor)

A   2M11   2M12  0M13  2 2 4 1

1

2

0 4 1

1

0

 26  24  0  20 36

Elementary Row Operations Elementary row operations are manipulations that can be performed on the row of a matrix. Elementary row operations is used to transform the augmented matrix of a system to a reduced / echelon form in order to solve the linear equation system

3 basics /legal elementary operations :

1. Interchange any two rows.

Ri  R j

 1 2  R1  R 2  3 4      1 2 3 4 37

Elementary Row Operations 2. Multiply any row by a non-zero scalar: c R i  1 2  2R 1  2 4      3 4 3 4 3. Add a multiple of one row to another row. 1 2   3 4

2R 1  R 2

5 8   3 4

Remark : illegal Row Operation Row cannot multiply with row

c Ri  R j 38

Example 5 Use elementary row operation to transform

1 3  1 0   into    2 4  0 1

Example 5…solution 1 3     2 4 

R2 : 2 3   1     2R 1 : 2 R 2  R 2  2 R1  0  10  0 1 3 R2 0 :    10  10 R 2  R 2   10  0 1 

R 1  R 1  3R 2

1 0   0 1

0 R1 : 1

4 6  10  10  10 1 3

  3R2 : 0

Inverse Matrix If A is square matrix and we can find another matrix of the same size B, such that

A B  B A  In

Then we call A invertible and we say that B is an inverse of the matrix A 1

1

A A  A A  In 41

For a square matrix A, the inverse is written A-1. When A is multiplied by A-1 the result is the identity matrix I. Non-square matrices do not have inverse.es.

NOTE: • Not all square matrices have inverses. • A square matrix which has an inverse is called invertible or nonsingular • Square matrix without an inverse is called noninvertible or singular. 42

Remarks on Inverse Matrix • • • •

Inverse matrix is only for square matrix The inverse of matrix A is denoted as A-1 Not all square matrix has inverse If det (A)=0, matrix A has no inverse, then A is a singular matrix. • If A is invertible, then A is a non-singular matrix 43

Finding Inverse Matrix “Using Adjoint Cofactor”

Case 1 : Inverse for 2×2 matrix a b  If matrix A   c d

is invertible, its inverse will be 1 A  A 1

 d b     c a  44

Example 6 Determine the inverse for the following matrix by using adjoint cofactor method.

 3 5   A   4 2 

1  2 5  A    26   4 3  1

 131   2   13

5 26 3 26

   45

Finding Inverse Matrix Case 2 :Inverse matrix for (3x3),(4x4), (5x5)….. Method 1 : Using Row Operation   A 33  

   

elementary row operation

  I3  

 A   1

Method 2 : Adjoint method (Transpose cofactor)

1 1 A   adjoint A   cofactor A  T A A 1

Finding Inverse Matrix “Using Row Operation”

Inverse for a 3×3 matrix.   A 33  

   

elementary row operation

  I3  

1

   

Find row operations that will convert the first 3 columns into I3. The last three columns should then contain A-1. 47

Example 7 Determine the inverse of matrix B by using row operation. 2  B  4 0 

0 3 3

5   0   4 

Please refer to Attachment 1 Please refer to ATTACHMENT 1-inverse matrix 3by3 using row operations.pdf 48

Finding Inverse Matrix “Using Adjoint Cofactor”

Inverse for a 3×3 matrix.  a11 a12 a13    If matrix A   a21 a22 a23  a  a a 32 33   31 is invertible, its inverse will be 1 1 A   adjoint A   cofactor A  T A A 1

Determine the inverse for the following matrix by using adjoint cofactor method.

Determine the inverse of matrix B. 2  B  4 0 

0 3 3

5   0   4 

Please refer to Attachment Please refer to ATTACHMENT -inverse matrix 3by3 adjoint cofactor.pdf 50

Success depends upon previous preparation, and without such preparation there is sure to be failure.