MATHEMATICS CHAPTER 4: MATRICES AND SYSTEMS OF LINEAR EQUATIONS
Operations on matrices
If A, B and C are matrices of the same order and 0 is the zero matrix, then
1. A + B = B + A
2. A + (B + A) = (A + B) + C
3. A + 0 = 0 + A = A
4. A + (-A) = 0 = (-A) + A
Properties of scalar multiplication
1. If A and B are two matrices of the same order and k is a scalar, then k(A + B) = kA + kB
2. If A is any matrix and k1, k2 are scalars, then
i. (k1 + k2)A = k1A + k2A
ii. k1(k2A) = k2(k1A) = (k1k2)A
Properties of matrix multiplication
1. If A, B, C are m x n, n x k and m x k matrix respectively, then
A(B + C) = AB + AC
A(B + C) = AB + AC
2. If A is a zero matrix of order m x n, B is of order n x k, then AB = 0
3. AB is not equals to BA since matric multiplication is not commutative
4. If A is a is a square matrix of order n x n and I is an identity matrix of the same order, then AI = IA = A.
5. AB = 0 (Zero matrix) does not imply that A = 0 or B = 0
6. Let A be a square matrix of order n x n, then A2 is defined as AA
In general, Am = A . A . ...A where m is any positive integer
The law of exponents is valid for powers of matrices, that is
ApAq = Ap+q, (Ap)q = Apq for p > 0, q > 0
In general, Am = A . A . ...A where m is any positive integer
The law of exponents is valid for powers of matrices, that is
ApAq = Ap+q, (Ap)q = Apq for p > 0, q > 0
7. Let I be an identity matrix, then I = I2 = I3 = ... = In
Properties of transposition of matrix
1.(AT)T = A
2.(kA)T = kAT, k is a scalar
3.(A + B )T = AT + BT
4.(AB)T = BTAT
Determinant of matrices
Cofactor, cij = (-1)i+j Mij and Cij = Mij (if i + j is even) or -Mij (if i + j is odd) and
Mij is the minor of aij
Mij is the minor of aij
Inverse of a matrix
Adjoint A, adj A = [cij]T
A-1 exists only when |A| is not equals to 0 (i.e. A is non-singular)
A-1 = (1/|A|) adj A
System of Linear Equation in Three Unknowns
The system of equations a11x + a12y + a13z = b1, a21x + a22y + a23y = b2, a31x + a32y + a33z = b3 can be expressed as a matrix equation, AX = B
MATHEMATICS CHAPTER 3: SEQUENCE AND SERIES
Arithmetic Progression (AP)
Arithmetic sequence: a, a + d, a + 2d, a + 3d, ..., a + (n-2)d, a + (n-1)d
The nth term, Tn = a + (n-1)d
Sum of nth term,
Sn = (n/2)[2a + (n-1)d]
OR
Sn = (n/2)(a + l)
*where l is the last term or l = a + (n-1)d
Geometric Progression (GP)
Geometris sequence: one in which the ratio (common ratio) of any term to its proceeding term is always the same
The nth term, Tn = arn-1
Sum of nth term,
When |r|<1, Sn = [a(1-rn)/(r-1)]
When |r|>1, Sn = [a(rn-1)/(r-1)]
Sum to infinity, Sinfinity = a/(1-r)
Binomial Theorem
Pascal's Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 5 1
Binomial coefficient, (nr) = n! / [r!(n-r)!]
Binomial theorem,
(a + b)n = (n0)anb0 + (n1)an-1b1 + (n2)an-2b2 + ... + (nr)an-rbr + ... + (nn)a0bn
OR
(a + b)n = the sum of (nr) an-rbr
The (r + 1)th term of the expansion of (a + b)n is denoted by Tr+1, and called the general term, Tr+1 = (nr)an-rbr
MATHEMATICS CHAPTER 2: EQUATIONS, INEQUALITIES AND ABSOLUTE VALUES
Properties of Absolute Values
1. |a| > 0
2. |a| = |-a|
3. |a+b| = |b+a|
4. |a-b| = |b-a|
5. |ab| = |a||b|
6. |a/b| = |a|/|b|
MATHEMATICS CHAPTER 1: NUMBER SYSTEM
Converting rational numbers as a ratio
Example; Write 2.135... as a ratio of 2 integers
Solution; Let x = 2.135...
x = 2.135135135... - (1)
1000x = 2135.135135... - (2)
(2) - (1); 999x = 2133
x = 79/37
Complex Numbers (a + bi)
i = i
i2= -1
Conjugate
· (a + bi) is (a - bi)
· Used to change division of complex number into complex number only
Rules of Indices
1. am x an = a(m+n)
2. am / an = a(m-n)
3. (am)n = amn
4. (ab)m = am . bm
5. a0 = 1
6. a-n = 1/an,
Rules of Logarithms
1. loga(mn) = logam + logan
2. loga(m/n) = logam - logan
3. logamn = nlogam
4. logam = (logbm) / (logba)
5. loga1 = 0
6. logaa = 1
7. alogan = n
