Definitions [2]
Definition: Matrix
A matrix is a rectangular arrangement of numbers arranged in rows and columns, enclosed in brackets [ ] or parentheses ( ).
Elements (Entries) of a Matrix
- Each number in a matrix is called an element (or entry).
Rows and Columns
- Horizontal lines → rows
- Vertical lines → columns
Order of a Matrix
- Order = number of rows × number of columns
- Written as m × n and read as “m by n”
Definition: Transpose of a Matrix
The transpose of a matrix is obtained by interchanging its rows and columns.
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If a matrix is A, its transpose is denoted by AT
-
If A is of order m × n, then
AT is of order n × m - First row of A becomes first column of AT, and so on.
Key Points
Key Points: Properties of Matrix Addition
| Property | Rule / Formula |
|---|---|
| Same Order Rule | Matrices can be added or subtracted only if they are of the same order |
| Commutative Property | A + B = B + A |
| Associative Property | A + (B + C) = (A + B) + C |
| Additive Identity | A + 0 = 0 + A = A |
| Additive Inverse | (A + (-A) = (-A) + A = 0 |
| Subtraction Rule | A - B = A + (-B) |
Key Points: Properties of Matrix Multiplication
| Property | Rule / Statement |
|---|---|
| Compatibility Rule | Matrices A and B can be multiplied only if the columns of A = the rows of B |
| Order of Product | If A is m × n and B is n × p, then AB is m × p |
| Non-Commutative | AB `\cancel(=)` BA (in general) |
| Associative Property | A(BC) = (AB)C |
| Distributive over Addition | A(B + C) = AB + AC |
| Zero Matrix Property | The product of two non-zero matrices can be a zero matrix |
| Cancellation Law | If AB = AC, it does not imply B = C |
| Identity Matrix | AI = IA = A (orders compatible) |
Key Points: Types of Matrices
| Type of Matrix | Key Property |
|---|---|
| Row Matrix | Has only one row (1 × n) |
| Column Matrix | Has only one column (m × 1) |
| Square Matrix | Number of rows = number of columns (n × n) |
| Rectangular Matrix | Number of rows ≠ , number of columns |
| Zero (Null) Matrix | All elements are 0 |
| Diagonal Matrix | Square matrix; all non-diagonal elements = 0 |
| Unit (Identity) Matrix | Diagonal matrix with all diagonal elements = 1 |
Concepts [20]
- Concept of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Matrix Addition
- Operation on Matrices
- Properties of Matrix Multiplication
- Types of Matrices
- Determinant of a Matrix
- Properties of Determinants
- Evaluation of Determinants
- Area of a Triangle Using Determinants
- Adjoint of a Matrix
- Inverse of a Square Matrix by the Adjoint Method
- Test of Consistency
- Applications of Determinants and Matrices
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Multiplication of Two Determinants
- Minors and Co-factors
- Some Special Cases of Matrix
- Rank of a Matrix
