F6 Mathematics T
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Revision Notes on Chapter 3 : Matrices Mat rices (Term (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 triangle 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).
A
6).
A B AT BT
7).
AB BT AT
8).
kA kAT
T
T
A T
T
T
(B) : Determinant of matrices
1).
8 2 3 4
8 4 2 3
2 3 1 2). If A 4 6 5 , 9 8 7 i j
3). Cofactor, Cij 1
M Miinor, M11
6 5
, M 32
8 7
2 1 4 5
, M 23
2 3 9 8
M ij , C11 M 11 , C32 M 32 , C 23 M 23
4). 2
1
3
4
0 1 2
1
2
3
0 1 2
-4 1
3
1
1
3
2
3
1
3
0 1
4 1 1
0
-2
3
3
2 3 1
2
3 3
4 1
4
0
1
2
2
1
1
2
-1
3
2 1 4
0
(C) : Properties of Determinants
1).
2).
1
2
3
1
4
5
6 2
7
8
9
3 6 9
1
2
3
4
4
5
6
7
8
9
4
2
7
5 8
5
,
(with interchanging rows & columns)
6
1 2 3 7
8
3).
9
1
2
3
1
2
3
4
5 6
1
4
1
2
5
2
3 6
3
2 rows identical
0
2 columns identical
4). ka1
ka2
ka3
ka1
a2
a3
a1
a2
a3
b1
b2
b3
kb1
b2
b3 k b1
b2
b3
c1
c2
c3
kc1
c2
c3
c2
c3
c1
6).
A AT
7).
AB A B if A & B are square matrices.
5).
1
2
3
4
5 6 0 3 4 0
0
0
0
0 1 2 0 5 6
(D) : Inverse Matrices
1).
1
A
1
2). If AB BA I , B A1 and A B 1
A
1
3).
a b d b 1 4). ad bc c a c d
AA1 A1 A I
5). When A 0, A1 not exist, A = singular matrix.
C11 C12 6). Adjoint Matrix, Adj A = C21 C22 C 31 C32 7). Inverse Matrix of A = A1 =
1 A
T
C13
C11 C21 C31 = C C23 12 C22 C32 C C33 13 C23 C33
adj A , if A 0
(E) : Using Elementary Row Operation to find
1). 3 operations :
A
-1
i ). Interchange any 2 rows. (e.g. R1 R2 ) ii ). Multiply a row by a scalar. (e.g. 2 R3 R3 ) iii). Multiply a row by a scalar and add to another row. (e.g. 2 R1 R3 R3 ) Note : Operation like 1 R3 R2 is not allowed as it totally eliminates all relations in row 2.
2). Steps for Elementary Row Operation: i ). Write the augmented matrix : A | I ii ). Use the operations above and change the augmented matrix into : iii). A1 B
I | B
3
3). Sequence guideline for Elementary Row Operation: Step 3 Step 1
Step 2
1 0 0 0 1 0 0 0 1
Step 6
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 x 3 y 6z 3 2 x 2 y 4 z 10 2 x 3 y z 7
AX = B
3 3 6 3 2). Augmented matrix = 2 2 4 10 | = A | B 2 3 1 7 3). Row-echelon form :
e.g. 1 :
1 a1 a2 a4 0 1 a a 3 5 0 0 1 a 6
e.g. 2 :
1 a1 a2 a3 a 5 0 0 1 a a 4 6 0 0 0 1 a 7
4). Reduce the augmented matrix to a matrix in row-eche lon form to find the values of the x, y & z of the system of linear equations or solve the equations. 5). Types of solutions: i ). If the row-echelon form has a row of ( 0 0 0 | c ) where c is a constant, then the system has no solution. ( |A| = 0 ) ii ). If the row-echelon form has a row of ( 0 0 0 | 0 ) , then the system has infinitely many solutions. ( |A| = 0 ) iii). If the row-echelon form does not have the form in (i) or (ii) above, then the system has a unique solution.
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(B) : Using Gaussian Elimination to solve a system of linear equations
Steps: 1). Write AX = B in matrix form from the 3 given linear equations to be solved. 2). Write the augmented matrix, ( A | B ).
1 0 0 1 3). Using Elementary Row Operation to reduce ( A | B ) to ( I | C ). (e.g. 0 1 0 1 ) 0 0 1 2 4). ( I | C ) is known as the Reduced row-echelon form. 5). Find the values of x, y & z.
(C) : Using the inverse of a matrix to solve a system of linear equations
Steps: 1). Write AX = B in matrix form from the 3 given linear equations to be solved. 2). Find A1 . 3). Use X A1B to find the values of x, y & z.
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