CSEC Mathematics
Section 9 – Matrices and Vectors (Vol. 2 Page 967 – R. Toolsie)
Matrices
A matrix is a rectangular array of numbers or letters consisting of ‘m’ rows and ‘n’ columns enclosed in a pair of curved or squared brackets and usually denoted by a capital letter.
Represent the Order of a Matrix in the Form m x n.
The order of a matrix is the number of rows by the number of columns. That is, m n or ‘m by n’. Each number or letter is called an
a 2 3 matrix, since it consists of 2 rows
and 3 columns. P = .
P = .
The matrix E = is called a square
matrix, because it consists of the same rows as columns. E is a 2 2 matrix.
The matrix F = is called a zero or
The matrix I = is called a unit
matrix. This matrix is the identity matrix for multiplication of 2 2 matrices. Any matrix multiplied by this matrix will result in the same matrix.
Equal Matrices
Two matrices are equal if they are of the same order and all the corresponding
elements are equal.
Given that A = and B = . If
A = B then = .
That is, a = e, b = f, c = g and d = h.
1. Given that = , find x and y.
Ans: x = – 2 y = 3
Addition and Subtraction of Matrices
Matrices of the same order can be added or subtracted, by adding or subtracting
corresponding elements.
Exercise
Given A = and B = , find:
a. A + B
b. B + A
c. B – A Ans:
d. A – B
Addition of Matrix is Commutative, that
Subtraction of matrices is not commutative, that is A – B B – A
Scalar Multiplication
Under scalar multiplication, each element in the pair of brackets is multiplied by the
scalar quantity or constant.
Exercise
Given A = and B = , find:
a. 3B b. 2A – B
c. A + 4B
In order to multiply matrices, we multiply the rows of the first matrix by the columns of the second matrix.
Given that A = and B = .
AB = =
BA = =
NOTE: A2 = A A = AA
Two matrices A and B are compatible or conformable for multiplication if A is an ‘m
n’ matrix and B is an ‘n p’ matrix. The resulting matrix will be an ‘m p’ matrix.
Multiplication is Associative
A(BC) = (AB)C
Hence, AB BA.
Exercise
State the result of the product of the following matrix multiplication.
1. 2 by 3 3 by 2 2. 3 by 1 1 by 2 3. 2 by 1 1 by 2 4. 1 by 3 3 by 2 5. 3 by 2 3 by 1 6. 2 by 1 2 by 1
Exercise
1. Given A = , B = and
a.i) AB ii) BA b. BC c. AC
d. (AB)C e. A(BC) f. B2
g. CA2
2.
3. Find the value of p and q if
.
Ans: p = 2 q = – 7
Given a 2 2 matrix A = . The determinant of the matrix A is:
det A or = ad – bc. That is, the product
of the leading diagonal elements – the product of the non-leading diagonal
elements.
Singular and Non-singular Matrix
A singular matrix is a matrix whose determinant is equal to zero. If ‘A’ is a singular matrix then
= = ad – bc = 0.
= = ad – bc 0.
Exercise
1. Find the determinant of the following 2
2 matrices and state whether the matrix is
singular or non-singular.
a. H = b. D =
c. J = d. M =
2. Find the value of k, given that the following matrices are singular matrix.
a) . Ans: k = - 6
b) Ans: k = 5
The adjoint of the matrix A = is: A
adjoint = . The adjoint is obtained by
interchanging the leading diagonal elements and multiply the non-leading diagonal
elements by – 1.
Exercise
State the adjoint of the following 2 2 matrices.
1. G = 2. K =
3. L = 4. T =
The inverse of a non-singular matrix
A = is:
A– 1 = = A adjoint =
. We can only find the inverse
of a non-singular matrix. If the determinant of a matrix A is zero, that is,
= = ad – bc = 0, then the inverse of
A does not exist.
Exercise
1. A = 2. H =
3. R = 4. W=
Using Matrix Method to Solve
Simultaneous Equations (Vol. 2 Page 1092 – R. Toolsie)
A pair of simultaneous equations can be solved by writing it as a matrix equation in the form
AX = B then X = A-1B which is used to
solve for the unknown in the equations.
Note: I = is the identity matrix for multiplication.
A – 1 A = I and AI = A or IA = A
D – 1 D = I and DI = D or ID = D
Example
Express each of the following pair of
equations in matrix form, that is AX = B. Then solve the following pair of equations using matrix method.
i. Ans: x = 1 & y = 2
ii Ans: x = 5 & y = 3 Exercise
b. Ans: x = 3 & y = 2
c. Ans: x = 3 & y = 7
d. Ans: x = 4 & y = 1
e. Ans: x = 2 & y = – 1
f. Ans: x = 2 & y = 3
Exercise
1. JANUARY 2016 – Question 11 b, c
a.
b.
3. JANUARY 2015 – Question 11a
4. MAY 2014 – Question 11a, b
5. JANUARY 2014 – Question 11a
6. MAY 2013 – Question 11b
Given M = .
a. Evaluate M – 1 , the inverse of M.
[2 marks]
b. Show that M – 1M = I. [2
marks]
c. Use a matrix method to solve for r, s, t
and u in the equation =
[5 marks]
a. The matrix J = represents a single transformation.
The image of the point P under
transformation J is (5, 4). Determine the coordinates of P.
[3 marks] b. Write down a matrix, H, of size 2 x 2, which represents an enlargement of scale factor 3 about
the origin. [1 mark]
c. Determine the coordinates of the point (5, – 7) under the combine transformation, H followed
8.
a.
b.
9.
10.
12.
13.
14.
16.
17.
Vectors
Physical quantities are divided into two types, vectors and scalar quantities.
A scalar quantity has magnitude (size) but no direction. For example, mass, length, volume, speed, time, distance and density.
A vector quantity has both magnitude and direction. For example, displacement,
weight, velocity, acceleration and force.
Note: Mass is a part of the magnitude of weight.
Speed is the magnitude of velocity.
Representing A Vector Quantity
A vector is a 2 1 matrix. It can be
represented by a straight line, with a length directly proportional to the magnitude of the quantity. An arrow is used to indicate the direction of movement of the vector.
Vector
The vector can be represented in column
matrix or column vector, . Thus, = ,
where x represents the horizontal movement
B A
and y represents the vertical movement. If x
is positive we move horizontally to the right
or if x is negative we move horizontally to
the left that number of units. If y is positive we move vertically up or if y is negative, we move that number of units vertically
downward.
A single common letter can also be used to
denote a vector. For example, a = .
= a =
Inverse (Negative) Vectors
Scalar Multiple of a Vector
Parallel Vectors
Two vectors are parallel if one vector is a scalar multiple of the other vector. In the diagram above, AB is parallel to CD since they are a scalar multiple of each other. In addition, two vectors are parallel if the
magnitude of one vector is a scalar multiple
C
B A
=
= D
Collinear Points
If two or more points lie on a straight line, we say the points are collinear.
Given that A, B and C lie on the same line and k, m, t are real numbers, then any one of the following must exists:
= k
= m
= t
Free Vectors (Displacement Vectors) and Position Vectors
A B
A free vector is not drawn with respect to the origin. Its starting point is not at the origin.
Exercise A
B
P
Q
K L
X
Write down the column vector for each free vector above.
a. b. c. d.
Position Vectors
A position vector is a vector which starts at
the origin. The vector is the
displacement of D from the origin O. is
called the position vector of D.
Any point (x, y) can be represented by a
Exercise
1. Use the graph above to write down the coordinates of each point below.
i. B ii. M iii. R iv. T v. H
2. Write down the column vector for each position vector.
i. ii. iii. iv.
v. - y x - x 0 B 1 2 3
Magnitude or Length of a Vector
The length of a vector , =
(using Pythagoras’ theorem). The magnitude or length of a vector is always positive.
tan = and = tan – 1 where
= the angle between and the x-axis.
Unit Vector
A unit vector is a vector whose magnitude (or modulus) is one unit. For example,
and are unit vectors with length one unit.
Activity – Find the magnitude or modulus
Triangle Law of Vectors
If two vectors are both going in a clockwise direction or both going in a
counterclockwise direction, then we can determine the sum of the two vectors by completing a triangle. The sum of the two vectors is called the resultant vectors.
Exercise
1. Determine the resultant of the following vectors.
a.
M H
b.
2. Draw a diagram for each of the
following, then find the resultant vector.
a. and
b. and
c. and
d. and
e. and
Exercise
4. MAY 2014 – Question 11b
7. JANUARY 2013 – Question 11a
The diagram below, not drawn to scale,
shows a parallelogram OKLM where O is the origin. The point S is on KM such that
. and .
Express each of the following in terms of u
and v.
a. [1 mark]
b. [2 marks]
c. [2 marks]
8.
10.
11.
