term 1 chapter 3 - matrices_new_2013

8/10/2019 Term 1 Chapter 3 - Matrices_new_2013

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F6 Mathematics T 1 of 4

Revision Notes on Chapter 3 : Matrices (Term 1)Name : ______________________________ Date : __________________

3.1: Matrices

(A) : Basic of Matrices

1). Be familiar with the following :

Null or zero matrix, diagonal matrix, identity matrix, symmetric matrix, row and column matrix,

upper triangular matrix, lower triangular matrix, equal matrix, order of matrix & , , , of matrices

with order up to 3 3.

2). A BC AB C (Associative)

3). A B C AB AC (Distributive over addition)

4). AB BA (Not commutative)

5). T

TA A

6). T T T

A B A B

7). T T TAB B A

8). T TkA kA

(B) : Determinant of matrices

1). 8 2

8 4 2 33 4

2). If 11 32 23

2 3 16 5 2 1 2 3

4 6 5 , Minor, M , M , M8 7 4 5 9 8

9 8 7

A

3). Cofactor,

11 11 32 32 23 231 , , ,

i j

ij ijC M C M C M C M

4).

2 1 30 1 4 1 4 0

4 0 1 2 1 32 3 1 3 1 2

1 2 3

1 3 2 3 2 1 -4 0 -1

2 3 1 3 1 2

1 3 2 3 2 1 1 - 2 30 1 4 1 4 0


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(C) : Properties of Determinants 2 of 4

1).

1 2 3 1 4 7

4 5 6 2 5 8 , or (with interchanging rows & columns)

7 8 9 3 6 9

TA A

2).

1 2 3 4 5 6

4 5 6 1 2 37 8 9 7 8 9

3).

1 2 3 1 4 1

1 2 3 2 5 2 04 5 6 3 6 3

(2 rows or columns are interchanged ) 2 rows identical 2 columns identical

4).

1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3

ka ka ka ka a a a a a

b b b kb b b k b b b

c c c kc c c c c c

5).

1 2 3 0 1 2

4 5 6 0 3 4 0

0 0 0 0 5 6

6). AB A B if A& Bare square matrices. 7).For a 3x3 matrix, 3kA k A ; k=constant.

(e.g. F ind 1

3A

in terms of A. H in ts: 1

3A

3A = 1

3 3A A

, 1

3A

=3

I

A=

1

27 A. )

(D) : Inverse Matrices

1). 11

AA

2). If 1 1, andAB BA I B A A B

3). 1 1AA A A I 4).

1

1a b d b

c d c aad bc

5). When 10,A A not exist, A= singular matrix.

6). Adjoint Matrix, Adj A=

11 12 13 11 21 31

21 22 23 12 22 32

31 32 33 13 23 33

=

T

C C C C C C

C C C C C C

C C C C C C

7). Inverse Matrix of a 3 x 3 matrix1 1

= = , if 0A A adj A AA

(E) : Using Elementary Row Operation to find A -1

1). 3 operations : i ). Interchange any 2 rows. (e.g.1 2

R R )

ii ). Multiply a row by a scalar. (e.g. 3 32R R )

iii). Multiply a row by a scalar and add to another row. (e.g. 1 3 32 R R R )Note :

Operation like 3 21 R R is not allowed as it totally eliminates all relations in row 2.

2). Steps for Elementary Row Operation to find an inverse matrix:

i ). Write the augmented matrix : |A I ii ). Use the operations above and change the augmented matrix into : |I B

iii). 1A B


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3). Sequence guideline for Elementary Row Operation to find an inverse matrix:

Step 3

Step 1

Step 2

1 0 0

0 1 0

0 0 1

Step 6

reduced row echelon form Step 5

Step 4

3.2: Systems of Linear Equations

(A) : Augmented Matrix, Row-echelon Form & Types of Solutions

1). System of linear equations :

3 3 6 3 2 2 4 10 AX = B

2 3 7

x y zx y z

x y z

2). Augmented matrix to solve a system of linear equations =

3). Row echelon form & Reduced row echelon form :

A matrix is in the row echelon formif the following conditions are met:

i ). All non-zero rows are above any row of all zeros.

ii ).Each leading entry (left most non-zero entry) of a row is in a column to the right of theleading entry of the row above it.

iii). All entries in a column below a leading entry are zeros.

e.g. 1 :

1 2 3 7

4 5 8

6 9

0

0 0

a a a a

a a a

a a

e.g. 2 :

1 2 3 4 8

5 6 9

7 10

0 0

0 0 0

a a a a a

a a a

a a

A matrix is in the reduced row echelon formif the following conditions are met:

i ). The conditions (i), (ii) & (iii) for the row echelon form are met, and

ii ). The leading entry in each non-zero row is 1.

iii). Each leading 1 is the only non-zero entry in its column.

e.g. :

1

2

3

1 0 0

0 1 0

0 0 1

a

a

a

4). Reduce the augmented matrix to a matrix in row-echelon form to find the values of the x, y & z ofthe system of linear equations or solve the equations.

3 3 6 3

2 2 4 10 = |

2 3 1 7

A B


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term 1 chapter 3 - matrices_new_2013

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