Tag: Matrices

Matrices is the plural of a matrix. A matrix is a rectangular array of numbers whose value and position in the arrangement is significant.

It is common to store large quantities of data and numerical information in form of tables.

converting tables to matrices

The table below shows marks scored by three students in an exam.

Name	Mathematics	Physics	chemistry
Jane	40	59	48
Phyllis	56	63	65
James	78	81	71

the data can be written as matrices form as :

consider the table below that shows English soccer results. A win scores 3 points, draw 1 point while loss wins zero points.

Team	matches played	wins	Draws	loses	Goals	points
Arsenal	25	18	5	2	49	59
Man city	25	15	5	5	51	50
Aston Villa	25	14	5	6	36	47
Man U	24	13	6	5	46	45
chelsea	23	12	5	6	45	41
Liverpool	23	11	7	5	40	40
Brentford	22	11	8	3	39	41
Sunderland	22	10	9	3	27	39
Fullham	21	9	4	8	35	31
Newcastle	19	8	5	5	35	29

with time, we may get to know what each team is scoring and remember each column values. In that case, we only need to remember the patterns only. The numerical data will thus be presented in form of a matrix as shown.

We have just converted a table to a matrix

A matix is represented as a brackets or square brackets as shown:

Capital letters in bold are commonly used to denote a matrix.

Each number in a matrix is referred to as an element of a matrix.

Order of matrices

Matrix size and shape depends on the number of rows and columns it has. If the number of rows equals number of columns, then it is a square matrix, otherwise it is a rectangular matrix.

A matrix with m rows and n columns is said to be a matrix of order m x n .

The order of matrix denotes the number of rows and columns in the matrix.

Therefore, a matrix of order 4 x 3 will have 4 rows and 3 columns.

A matrix of order 1 x n is called a row matrix.

A matrix of order m x 1 is said to be a column matrix.

A square matrix is a matrix of order m x m or n x n.

The matrix below is a square matrix of order 4 x 4:

positioning elements in matrices

Position of an element in a matrix is described by use of suffices.

If A is a matrix and a_m,n is an element in that matrix, then a_m,n is the element in the m^th row and on the n^th column.

consider the matrix above, 1 is an element at a_3,3 position.

summary

practice exercise on matrices

1. The table below shows marked scored in three tests by grade 10 students.

Student	Test 1	Test 2	Test 3
Jane	68	73	56
Milliam	79	71	67
Nancy	67	59	63
James	45	56	78
Mark	62	70	86
Eliud	71	74	82
Brian	89	84	87
Harriet	84	76	77

(a) Write down the information from the table into a matrix and state order of the matrix.

(b) Write test 2 as a column matrix and state it’s order

(c) write as a matrix, Brian’s scores in the three test

(d) change the marks in test 1 and test to to 60% of the current value.

(e) convert test 3 to 40% of the current value.

(f) make a new matrix by adding a fourth column in the matrix above with the totals of (d) and (e) above

2. Given matrix A below. state the elements in:

(a) a_1,3 (b) a_2,4 (c) a_3,2 (d)a_4,1

Related topics

• Introduction Arithmetic to series
• Matrix (mathematics)

Databases

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1. Introduction

Matrices are rectangular arrangements of numbers organized in rows and columns. They are powerful tools used to represent and manipulate data in science, economics, engineering, computer graphics, and real life decision-making. Operations on Matrices includes addition, subtraction, multiplication and division, just like arithmetic operations on numbers.

Example of a Matrix

A = | 2   4 |
    | 1   3 |

• This is a 2 × 2 matrix
• 2 rows, 2 columns

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2. Types of Matrices (Quick Review)

Type	Description
Row Matrix	Only one row
Column Matrix	Only one column
Square Matrix	Rows = Columns
Zero Matrix	All elements are zero
Identity Matrix	1s on the main diagonal

Identity Matrix (2 × 2)

I = | 1   0 |
    | 0   1 |

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Operations on Matrices

A. Addition of Matrices

Two matrices can be added only if they have the same order.

Rule:

Add corresponding elements.

A = | 2   3 |      B = | 1   4 |
    | 5   6 |          | 7   2 |

A + B = | 2+1   3+4 |
        | 5+7   6+2 |

      = | 3   7 |
        |12   8 |

📌 Real-life idea: Adding daily sales from two shops.

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B. Subtraction of Matrices

Same rule as addition, but subtract corresponding elements.

A − B = | 2−1   3−4 |
        | 5−7   6−2 |

      = | 1  −1 |
        |−2   4 |

📌 Real-life idea: Comparing profits between two months.

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C. Scalar Multiplication

Multiply each element by a number (scalar).

Let A = | 2   1 |
        | 3   4 |

3A = | 6   3 |
     | 9  12 |

📌 Real-life idea: Increasing production costs by a fixed rate.

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D. Multiplication of Matrices

Matrix multiplication is not commutative:

AB ≠ BA (in most cases)

For matrix multiplication, the following conditions need to be met.

1. Number of columns in first matrix = number of rows in second matrix.

A = | 1  2 |      B = | 3 |
    | 4  5 |          | 6 |

AB = | (1×3 + 2×6) |
     | (4×3 + 5×6) |

   = | 15 |
     | 42 |

📌 Real-life idea: Calculating total cost using quantity × price.

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4. Real-Life Applications of Matrices

1. Economics & Business

Matrices are used to calculate profit, loss, production costs, and wages.

Cost Matrix × Quantity Matrix = Total Cost

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2. Computer Graphics

Matrices are used to rotate, enlarge, and shrink images in video games and animation.

[Image Points] × [Transformation Matrix]

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3. Transportation

Used to organize routes, distances, and schedules between cities.

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4. Science & Engineering

Matrices model electrical circuits, population growth, and chemical reactions.

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5. Worked Examples

Simple Example

Add:

A = | 2  1 |
    | 3  4 |

B = | 5  6 |
    | 1  2 |

Solution:

A + B = | 7  7 |
        | 4  6 |

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Complex Example

Multiply:

A = | 1  2  3 |
    | 4  5  6 |

B = | 1  2 |
    | 3  4 |
    | 5  6 |

Solution:

AB = | (1×1+2×3+3×5)   (1×2+2×4+3×6) |
     | (4×1+5×3+6×5)   (4×2+5×4+6×6) |

   = | 22   28 |
     | 49   64 |

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6. Revision Exercise

(Arranged from simple to complex)

A. Basic Level

1. Add:

| 3  2 |   +   | 1  4 |
| 5  1 |       | 2  3 |

2. Subtract:

| 7  5 |   −   | 4  2 |
| 6  3 |       | 1  1 |

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B. Intermediate Level

3. If

A = | 2  3 |
    | 1  4 |

Find 3A.

4. Determine whether the following multiplication is possible:

(2 × 3) × (2 × 2)

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C. Advanced Level

5. Multiply:

| 1  0  2 |   ×   | 3 |
| 2  1  1 |       | 1 |
                  | 2 |

6. If

A = | 1  2 |
    | 3  4 |

Find A².

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7. Application Questions (Real Life)

1. A shop sells two products. The cost matrix is

| 50  30 |

and the quantity matrix is

| 10 |
| 20 |

Find the total cost.

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2. A school records boys and girls in two classes using matrices. How can matrix addition help find the total number of students?

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3. A graphics designer enlarges an image using a transformation matrix. Explain how scalar multiplication is applied.

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4. A transport company uses a matrix to represent distances between cities. How can matrix multiplication help calculate total travel distances?

Related topics

• Introduction to matrices
• Gradients of line
  • Matrices Basic Operations.pdf

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