1. Introduction to Matrices
A matrix is an important mathematical tool used to arrange numbers or algebraic expressions systematically. Matrices are useful in engineering calculations, solving simultaneous linear equations, computer graphics, image processing, circuit analysis, data science and mathematical modelling.
A matrix helps represent a collection of values in rows and columns.
2. Definition of a Matrix
Definition: A matrix is a rectangular arrangement of numbers, symbols or algebraic expressions in rows and columns, enclosed within brackets.
The numbers or expressions inside a matrix are called its elements or entries. Matrices are generally denoted by capital letters such as A, B, C, P and Q.
A = [ 1 2 3 ]
[ 4 5 6 ]
This matrix has 2 rows, 3 columns and 6 elements.
3. General Representation of a Matrix
A matrix having m rows and n columns is represented by:
A = [ a₁₁ a₁₂ … a₁ₙ ]
[ a₂₁ a₂₂ … a₂ₙ ]
[ ⋮ ⋮ ⋱ ⋮ ]
[ aₘ₁ aₘ₂ … aₘₙ ]
Here:
- A = the matrix
- m = number of rows
- n = number of columns
- aᵢⱼ = element in the i-th row and j-th column
General element: aᵢⱼ (row first, column second)
4. Order or Size of a Matrix
The order of a matrix is the number of rows multiplied by the number of columns, written as:
Order = m × n
The first number indicates rows and the second number indicates columns.
Example 1
A = [ 2 4 6 ]
[ 1 3 5 ]
Rows = 2; Columns = 3; therefore, order(A) = 2 × 3.
Example 2
B = [ 1 2 ]
[ 3 4 ]
[ 5 6 ]
[ 7 8 ]
Rows = 4; Columns = 2; therefore, order(B) = 4 × 2.
Important: Always write the order as rows × columns, not columns × rows.
5. Elements of a Matrix
Consider the matrix:
A = [ 10 20 30 ]
[ 40 50 60 ]
[ 70 80 90 ]
ElementValuePositiona₁₁101st row, 1st columna₁₂201st row, 2nd columna₂₃602nd row, 3rd columna₃₁703rd row, 1st columna₃₃903rd row, 3rd column
For example, a₂₃ means the element in the second row and third column.
6. Types of Matrices Based on Order1. Row Matrix
A matrix containing only one row.
[ 2 4 6 8 ]
Order: 1 × 4
2. Column Matrix
A matrix containing only one column.
[ 3 ] [ 5 ] [ 7 ]
Order: 3 × 1
3. Rectangular Matrix
A matrix in which the number of rows is not equal to the number of columns.
[ 1 2 3 ] [ 4 5 6 ]
Order: 2 × 3
4. Square Matrix
A matrix in which the number of rows equals the number of columns.
[ 1 2 ] [ 3 4 ]
Order: 2 × 2
7. Important Terms Related to MatricesTermMeaningElementA number or expression inside a matrix.RowHorizontal arrangement of elements.ColumnVertical arrangement of elements.OrderNumber of rows × number of columns.Square matrixA matrix with equal numbers of rows and columns.Main diagonalEntries from top-left to bottom-right in a square matrix.Diagonal elementsEntries a₁₁, a₂₂, a₃₃, … in a square matrix.Equal matricesMatrices of the same order with corresponding equal elements.8. Solved ExamplesExample 1: Find the order
A = [ 2 3 4 5 ]
[ 6 7 8 9 ]
[ 1 2 3 4 ]
There are 3 rows and 4 columns. Hence, order(A) = 3 × 4.
Example 2: Find a₂₃
A = [ 5 6 7 ]
[ 8 9 10 ]
[ 11 12 13 ]
The element in row 2, column 3 is 10. Therefore, a₂₃ = 10.
Example 3: Identify the matrix type
A = [ 4 0 0 ]
[ 0 5 0 ]
[ 0 0 6 ]
This is a 3 × 3 square diagonal matrix because all non-diagonal elements are zero.
9. Important Points
- A matrix is a rectangular arrangement of elements.
- Order of a matrix = m × n (rows × columns).
- A matrix with one row is a row matrix.
- A matrix with one column is a column matrix.
- A matrix with equal rows and columns is a square matrix.
- Total number of elements in an m × n matrix = mn.
- aᵢⱼ denotes the element in the i-th row and j-th column.
- Two matrices are equal only if their orders and corresponding elements are equal.
MEMORY TRICK: M × N = Rows × Columns
For aᵢⱼ, remember: row first, column second.