Week 4 – SS3 First Term Mathematics Notes

WEEK 4 DATE………..
Matrices and determinants: concept, the basic operations of matrices. Identity matrices and equal matrices
MATRICES
Matrix is a rectangular array of numbers or elements in a row or column.
e.g ( a b ), a

  b
Elements arranged along the horizontal are called ROW. While elements arranged along the vertical is called COLUMN.
E,g
5 7 5 7 row 1
8 12 8 12 row 2
Column 1 2
NOTATION: A matrix is denoted by capital letter and the elements by small letters with reference to the position of the element. The position is defined in terms of the number of rows and columns.The first indicating the row, the second the column, thus:

a₁₁ a₁₂ a₁₃b₁₁ b₁₂
A = a₂₁ a₂₂ a₂₃B = b₂₁b₂₂
a₃₁ a₃₂ a₃₃
Hence, a₂₁indicates the element in the second row and first column.
EVALUATION: Given the matrix,C= 6 -5 1 -3 write out the elements occupying the following,
Positions.C₁₁, C_21, C₃₂, C₄₂, C₄₄ andC₃₄ 2 -4 8 3
4 -7 -6 5
-2 9 7 -1
Order of a matrix: A matrix can be identified or described by its order. In describing a matrix, the number of rows is stated first before the number of columns.
E.g 6 2 8 is a 2 x 3 matrix, i.e. order 2 by 3.
5 7 3

 BASIC OPERATION OF MATRICES:
Addition and subtraction of matrices: Two or more matrices can be added or subtracted when they are of the same order e.g 2 x 2, 3 x 3 and so on. The sum or subtraction is then determined by adding or subtracting corresponding elements.

If A = a₁₁ a₁₂ B = b₁₁ b₁₂A + B = a₁₁ + b₁₁ a₁₂ + b₁₂ A – B = a₁₁ –b₁₁ a₁₂ –b₁₂

  a₂₁ a₂₂ b₂₁ b₂₂ a₂₁+ b₂₁ a₂₂ + b₂₂a₂₁– b₂₁ a₂₂ – b₂₂

 
 Example: Given the matrices below, find I A + B II A – B III B – A .

  A = 7 6 5 B = 12 -6 4
9 4 8 2 10 1

I A + B = 19 0 9 II A – B = -5 12 1 III B – A = 5 -12 -1
11 14 9 7 -6 7 -7 6 -7

 Addition of matrices is commutative, i.e A + B = B + A but matrix subtraction is not commutative,
A – B ≠ B – A

 MULTIPLICATION OF MATRICES:

1. Scalar multiplication: This is the multiplication of a matrix by a single number and it is done by multiplying each element in the matrix by the scalar.

e.g If C = 3 -8 12 find I 2C II -3C
4 5 7

I 2C = 6 -16 24 II – 3C = -9 24 -36
8 10 14 -12 -15 -21

1. Multiplication of two matrices: Two matrices can be multiplied together only when the number of columns in the first is equal to the number of rows in the second matrix.

If A= a₁₁ a₁₂ B = b₁then A. B = a₁₁b₁ + a₁₂b₂
a₂₁ a₂₂ b₂a₂₁b₁ + a₂₂b₂

  Matrix by matrix multiplication is not commutative, A.B ≠ BA

 Example: Given the matrices A = 2 3 B = 4 1 find AB.
5 7 8 9
2 3 4 1
AB = x
5 7 8 9

  AB = 2×4 + 3×8 2 x1 + 3×9 8+24 2 +27
=
5×4 + 7×8 5×1 + 7×9 20 +56 5+63

  AB = 32 29

1. 69

 
 
EVALUATION: 1. Find AC given the matrices A = -3 2 5 C = 4 7 2
0 1 8 3 -5 1
9 0 4
2. Given that A = 12 -8 B = 1 4 and C = 9 -5
3 6 2 8 15 10
Find I. A + B II. B – C III. 2A – B + 3C

 TYPES OF MATRICES:
Equal matrices: Two matrices are said to be equal if corresponding elements are equal and the matrices are of the same order.
Square matrix: Is a matrix having the same number of rows and columns. E.g 2 x 2, 3 x 3,and so on
Diagonal matrix: This is a square matrix with all elements zero except those on the main diagonal.
2 0 0
0 5 0
0 0 8

 Identity matrix: It is also called unit matrix and is a diagonal matrix in which the elements on the main diagonal are equal to one (1). It is denoted by I.
2 x 2 ,I =1 0 3 x 3, I = 1 0 0
0 1 0 1 0
0 0 1
Null matrix: Is a matrix whose elements are zero. It is denoted by o. i.e
0 0 0
0 0 0
0 0 0

 TRANSPOSE OF A MATRIX: The matrix obtained by interchanging the rows and columns of a matrix is called the transpose matrix. If B is the original matrix, its transpose is denoted by B^T.

If A = 2 5 , then A^T = 2 -8 7
-8 3 5 3 4
7 4
EVALUATION: If A = 1 2 3 and B = 7 10 find I A.B and (A.B)^T
4 5 6 8 11
9 12

 
 WEEKEND ASSIGNMENT
Given that A = 4 2 3 B = 1 8 9
5 7 6 3 5 4

1. Find A + B. A. 5 10 12 B. 3 6 12
8 12 10 2 -2 2

 2. Find A – B A. -3 -6 -6 B. 3 -6 -6
-2 2 -2 2 2 2

3. Find (A + 2B) ^T    A. 6 18 21 B. 6 11 C. 6 18
     11 17 14 18 17 21 11
     21 14 17 14
4. Find A² – 4I
5. Find BA.

 
 
 THEORY
1. Given that     P = 1 5 find 2p² – 3p + 5I
     -4 2

2.      1 3 2      2 4 – 3 3 1 6
If P = 8 -4 4      Q =      3 8 4 R = 4 3 2
7 3 5      -1 3 6 2 – 1 1

  Find (a) 5P + 2Q (b) 4Q – 2R (iii) 2P + Q + 3R (iv) PR

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Completing the square
To make a given expression a perfect square, the quantity to be added is the square of half of the coefficient of x ( or whatever letter is involved).
Examples:
In each of the following, add the term that makes the given expression into a perfect square , then write the result as the square of a bracketed expression.
1. g² – 4 ²/3 g
2. k² – ¹¹/3 k
3. m² + 3mn.
Solutions.
g² – 4 ²/₃ g
the coefficient of g is – ⁴²/3 = –¹⁴/3
half of –¹⁴/₃ = ½ x –¹⁴/₃ =-⁷/₃
Square of half of coefficient of g = (-⁷/3)² = 49 ( + 5⁴/₉)
9
:. 49 must be added to the given expression to make it a perfect square
9
:. g² -4 ²/₃ + ⁴⁹/₉ = ( n – ⁷/₃)
k² – 1 ¹/₃ k. the coefficient of k is ……..¹¹/3 – –⁴/₃
Half of –⁴/₃ = ½ x –⁴/₃ = –²/₃
Square of half of coefficient of k = ( –²/₃)² = + ⁴/₉.
:. ⁴/9 must be added to the given expression to make it a perfect square
:. K² – 1 ¹/₃ k + ⁴/₉ = ( k – ²/₃)²

1. m ^{c xx} + 3mn

the coefficient of m is 3n .
half of + 3n =¹/₂ x + 3n = + 3n
2.
EVALUATION.
In each of the following add the term that makes the given expression into a perfect square . Write the result as the square of a bracketed expression
1b² – ⁴/5b
2.u² – 1 ³/5 u
WEEKEND ASSIGNMENT
1.Find the value of k such that x2 + 5x +k is a perfect square.
(a) 2 ½         (b) 4 ¼ ( c) 6 ¼ ( d) 25 (e) 100.
2. What must be addedto n² + 1 1/s n to make it a perfect square?
(a) ³/₅        (b) 1 ¹/₅ ( c) ¹⁶/ ₉     ( d) ⁹/₁₆ ( e ) 1 ¼
3. Solve the equation
( x + 1 ¼ )² = 1 ⁹/₁₆
( a) 2 ½, 0 ) (b) (0,2) ( c ) ( 0, – ) ( d ) ( 0, -2 ½ ) (e) ( 3, 1 ½ ).
4.Solve the equation
( x +¹/₃ )² =⁴/₉
( – ¹/₃, 1) (b) (1, ¹/₃ ) ( c) (2, ²/₃ ) (d ) (-²/₃, 3) (e )(¹/₃, -1).
5. What must be added to v2 – 3/4 v to make it a perfect square?
    (a) ⁹/₆₄        (b) ³/₈        ( c) –³/₈    (d) 7 ¹/₉ (e) ²/₉
Theory
In each of the following, add the term that makes the given expression into a perfect square. Then write the result as the square of a bracketed expression:

 ii. u² – 1 ³/5u
2. a² – 6ad

 
 TOPIC: SOLUTION OF QUADRATIC EQUATION & SYMMETRIC PROPERTIES OF THE ROOT OF QUATION EQUATION
CONTENT

• Method of Factorization
  
  • Completing the square method.
  • Quadratic formula
  • Sum & Product of Roots of a Quadratic Equation
  • Symmetric Properties of Roots

Method of Factorization
A quadratic equation is an expression of the form ax² + bx + c = 0 in which a, b & c are numerals; and also the highest power of x is 2 & that the power of x will neither be fractions nor negatives. Quadratic equations can be solved using the method of factorization, completing the square, quadratic formula& graphical method
Steps in solving quadratic equation: (1)examine the middle term whose power of x is 1. (2) Find the product of the first & last term. (3) Find two terms whose sum is equal to the middle term & product is equal to the value of the product of the first & last term (4) Replace the middle term by two the two terms in step 3. (5) Factorize the first two & last two terms (6) equate the linear factors to zero to find the value of x.
Example – Solve by factorization: X² + 7X + 10 = 0
Solution
X² + 7X + 10 = 0
X² +2X + 5X + 10 = 0
X(X + 2) + 5(X + 2) = 0
(X+2) (X + 5) = 0
X + 2 = 0
X = -2 OR
X + 5 = 0
X = -5
Hence X = -2 or -5