Vectors and Matrices - CSEC Mathematics

Introduction to Vectors

A vector is a quantity that has both magnitude (size) and direction. Vectors are used to represent physical quantities like displacement, velocity, and force.

Vector Representation

Vectors can be represented in several ways:

Vector AB A B

Vector Notation

Vectors are typically written with:

Vector Operations

Vector Addition

Vectors can be added using:

a b a + b a b a + b

Example: Vector Addition

If a = (3, 2) and b = (-1, 4), find a + b.

Solution:

a + b = (3 + (-1), (2 + 4)) = (2, 6)

Scalar Multiplication

Multiplying a vector by a scalar (real number) changes its magnitude but not its direction (unless the scalar is negative, which reverses direction).

If v = (x, y) and k is a scalar, then kv = (kx, ky)

Example: Scalar Multiplication

If v = (2, -3), find 4v.

Solution:

4v = (4×2, 4×-3) = (8, -12)

Magnitude of a Vector

The magnitude (length) of vector v = (x, y) is given by:

|v| = √(x² + y²)

Example: Finding Magnitude

Find the magnitude of v = (3, 4).

Solution:

|v| = √(3² + 4²) = √(9 + 16) = √25 = 5

Unit Vectors

A unit vector has magnitude 1. The unit vector in the direction of v is:

û = v/|v|

Example: Finding Unit Vector

Find the unit vector in the direction of v = (3, 4).

Solution:

|v| = 5 (from previous example)

û = (3/5, 4/5)

Introduction to Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to solve systems of equations, represent transformations, and more.

Matrix Notation

A matrix with m rows and n columns is called an m×n matrix. It is typically written with square brackets:

a
b
c
d

Types of Matrices

Matrix Type Description Example
Row matrix Single row [1 2 3]
Column matrix Single column
1
2
Square matrix Same number of rows and columns
1
2
3
4
Zero matrix All elements are zero
0
0
0
0

Matrix Operations

Matrix Addition

Matrices of the same size can be added by adding corresponding elements.

Example: Matrix Addition

Add

1
2
3
4
and
5
6
7
8

Solution:

6
8
10
12

Scalar Multiplication

A matrix can be multiplied by a scalar by multiplying each element by that scalar.

Example: Scalar Multiplication

Multiply

1
2
3
4
by 2

Solution:

2
4
6
8

Matrix Multiplication

Matrix multiplication is more complex. For matrices A (m×n) and B (n×p), the product AB is an m×p matrix where each element is the dot product of a row from A and a column from B.

Example: Matrix Multiplication

Multiply

1
2
3
4
by
5
6
7
8

Solution:

First row, first column: (1×5) + (2×7) = 5 + 14 = 19

First row, second column: (1×6) + (2×8) = 6 + 16 = 22

Second row, first column: (3×5) + (4×7) = 15 + 28 = 43

Second row, second column: (3×6) + (4×8) = 18 + 32 = 50

19
22
43
50

Important: Matrix multiplication is not commutative (AB ≠ BA in general).

Determinants and Inverses

Determinant of a 2×2 Matrix

For matrix A =

a
b
c
d
, the determinant is det(A) = ad - bc

Example: Finding Determinant

Find the determinant of

3
4
1
2

Solution:

det = (3×2) - (4×1) = 6 - 4 = 2

Inverse of a 2×2 Matrix

A matrix A has an inverse A⁻¹ if det(A) ≠ 0. The inverse is:

A⁻¹ = (1/det(A)) ×

d
-b
-c
a

Example: Finding Inverse

Find the inverse of

3
4
1
2

Solution:

From previous example, det = 2

A⁻¹ = (1/2) ×

2
-4
-1
3
=
1
-2
-0.5
1.5

Applications of Vectors and Matrices

Solving Systems of Equations

Matrices can solve systems like:

2x + 3y = 5

x - y = 1

This can be written as AX = B, where:

A =

2
3
1
-1
, X =
x
y
, B =
5
1

The solution is X = A⁻¹B

Transformations

Matrices can represent geometric transformations:

Glossary of Terms

Column Vector
A matrix with only one column.
Determinant
A scalar value that can be computed from a square matrix and encodes certain properties of the matrix.
Identity Matrix
A square matrix with 1s on the main diagonal and 0s elsewhere.
Inverse Matrix
A matrix that when multiplied by the original matrix gives the identity matrix.
Magnitude
The length or size of a vector.
Matrix
A rectangular array of numbers arranged in rows and columns.
Scalar
A single real number (as opposed to a vector or matrix).
Square Matrix
A matrix with the same number of rows and columns.
Unit Vector
A vector with magnitude 1.
Zero Matrix
A matrix where all elements are zero.

Self-Assessment Questions

  1. Find the sum of vectors (2, 5) and (-3, 1).

    2. (2 + (-3), 5 + 1) = (-1, 6)

  2. Calculate the magnitude of the vector (6, -8).

    3. √(6² + (-8)²) = √(36 + 64) = √100 = 10

  3. Find the determinant of the matrix
    4
    1
    3
    2

    4. (4×2) - (1×3) = 8 - 3 = 5

  4. Multiply the matrices
    1
    0
    2
    -1
    and
    3
    4

    5. First row: (1×3) + (0×4) = 3

    Second row: (2×3) + (-1×4) = 6 - 4 = 2

    3
    2
  5. Find the inverse of the matrix
    2
    1
    1
    1

    6. det = (2×1) - (1×1) = 1

    A⁻¹ =

    1
    -1
    -1
    2

  6. Find the unit vector in the direction of (3, -4).

    7. Magnitude = √(3² + (-4)²) = 5

    Unit vector = (3/5, -4/5)

  7. If A =
    1
    2
    3
    4
    and B =
    0
    -1
    2
    3
    , find AB.

    8. AB =

    (1×0)+(2×2)
    (1×-1)+(2×3)
    (3×0)+(4×2)
    (3×-1)+(4×3)
    =
    4
    5
    8
    9

  8. Find 3A - 2B where A = (1, -2) and B = (3, 0).

    9. 3A = (3, -6), 2B = (6, 0)

    3A - 2B = (3-6, -6-0) = (-3, -6)

  9. Does the matrix
    2
    4
    1
    2
    have an inverse? Why or why not?

    10. det = (2×2) - (4×1) = 4 - 4 = 0

    No inverse because the determinant is zero.

  10. Solve the system using matrices: 2x + y = 5, x - y = 1

    11. Matrix form:

    2
    1
    1
    -1
    x
    y
    =
    5
    1

    det = (2×-1) - (1×1) = -3

    Inverse =

    -1
    -1
    -1
    2
    × (1/-3)

    Solution: x = 2, y = 1