Definitions [5]
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:
\[A=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\mathrm{and} B=
\begin{bmatrix}
2 & & 3 \\
1 & & 5
\end{bmatrix}\]
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.
Skew-Symmetric Matrix: A square matrix A = [aij] n×n is skew-symmetric if Aᵀ = −A i.e., aij = −aji for all i and j.
A square matrix A = [aᵢⱼ]ₙ×ₙ is symmetric if Aᵀ = A
i.e., aᵢⱼ = aⱼᵢ for all i and j.
Theorems and Laws [1]
If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
If A and B are symmetric matrices.
∴ A’ = A and B’ = B
(AB – BA) = (AB)’ – (BA)’ ...[∵ (X – Y) = X’ – Y’]
= B’A’ – A’B’ ...[∵ (XY) = Y’X’]
= BA – AB ...[∵ B’ = B, A’ = A]
= –(AB – BA)
∴ AB – BA is a skew symmetric matrix.
Key Points
-
Matrix: A rectangular array of elements.
-
Element: An entry inside a matrix.
-
Order: Size of a matrix written as rows × columns.
-
Row: Horizontal set of elements.
-
Column: Vertical set of elements.
-
aij: Element in the i-th row and j-th column.
| Matrix Type | Order | Key Property |
|---|---|---|
| Row Matrix | 1 × n | Only one row |
| Column Matrix | m × 1 | Only one column |
| Square Matrix | n × n | Rows = Column |
| Rectangular Matrix | m × n (m ≠ n) | Rows ≠ Columns |
| Diagonal Matrix | n × n | Square; non-diagonal elements = 0 |
| Scalar Matrix | n × n | Diagonal; all diagonal elements equal |
| Identity Matrix | n × n | Scalar matrix with diagonal = 1 |
| Zero Matrix | Any order | All elements = 0 |
-
Equality of matrices is possible only when the order is the same.
-
Corresponding elements must be compared position by position.
-
If even one corresponding entry differs, the matrices are not equal.
