Definitions [4]
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”
Two matrices are equal if and only if:
- They have the same order (same number of rows and columns), and
- Their corresponding elements are equal.
Example:
\[A=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\mathrm{and} B=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\]
Two matrices are said to be comparable if they are of the same order, that is, they have the same number of rows and the same number of columns.
Example:
A = \[\begin{bmatrix}
2 & -3 & 0 \\
5 & 1 & -4
\end{bmatrix}\] \[ B =
\begin{bmatrix}
6 & 7 & 1 \\
8 & 0 & 9
\end{bmatrix}\]
The transpose of a matrix is obtained by interchanging its rows and columns.
-
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
| 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 |
| 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) |
| 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) |
