Comprehensive theory, key formulas, diagrams, and memory aids for Matrices.
A matrix is a rectangular arrangement of numbers (or functions) in rows and columns enclosed in square brackets. Matrices provide a compact and powerful notation for handling large collections of data and for solving systems of linear equations. The German mathematician Arthur Cayley is regarded as the father of matrix theory, and today matrices appear in physics, economics, computer graphics, engineering, and statistics.
This chapter introduces the notation of a matrix, the order (dimension) of a matrix, and the different types of matrices such as row, column, square, diagonal, scalar, identity, zero, and triangular matrices. It then develops matrix equality, the arithmetic of matrices including addition, scalar multiplication, and matrix multiplication, and the special properties and algebraic rules these operations obey. The chapter closes with the transpose of a matrix, symmetric and skew-symmetric matrices, and elementary row and column operations.
Matrices are important not merely as an abstract structure but as a computational tool. The operations developed here, especially matrix multiplication and the transpose, will be used in the next chapter on determinants, in solving linear equations by the matrix method, and in understanding the geometric transformations in three-dimensional geometry. A careful study of when matrix multiplication is defined, and why AB need not equal BA, is the single most important skill in this chapter.
A matrix with m rows and n columns is said to be of order m x n (read as m by n). It is generally written as
$$A = [a_{ij}]_{m \times n}$$
where $a_{ij}$ denotes the entry in the i-th row and j-th column. The element $a_{ij}$ is located at the intersection of the i-th row and j-th column.
For example, the matrix $$A = \begin{bmatrix} 2 & 5 & -1 \ 3 & 0 & 4 \end{bmatrix}$$ is of order 2 x 3. Here $a_{11} = 2$, $a_{12} = 5$, $a_{13} = -1$, $a_{21} = 3$, $a_{22} = 0$, and $a_{23} = 4$.
A matrix is a rectangular array only; it is not a number, and its determinant (defined in the next chapter) exists only for square matrices.
Two matrices A and B are equal if and only if they have the same order and $a_{ij} = b_{ij}$ for every corresponding pair of entries. Matrices of different orders can never be equal, no matter what their entries are.
Two matrices A and B can be added only when they have the same order. Then $A + B = [a_{ij} + b_{ij}]$, adding corresponding entries.
If k is a scalar (number), then $kA = [ka_{ij}]$; every entry is multiplied by k. The negative of a matrix is $-A = [-a_{ij}]$, and subtraction is defined as $A - B = A + (-B)$.
Two matrices A and B can be multiplied as AB only if the number of columns of A equals the number of rows of B. If A is of order m x n and B is of order n x p, then AB is of order m x p, and
$$(AB){ij} = \sum{k=1}^{n} a_{ik} b_{kj}$$
The entry in the i-th row and j-th column of AB is the dot product of the i-th row of A with the j-th column of B.
If A is a matrix of order m x n, its transpose $A^T$ (or A') is the matrix of order n x m obtained by interchanging rows and columns: $(A^T){ij} = a{ji}$.
$$(A + B)^T = A^T + B^T, \quad (kA)^T = kA^T, \quad (AB)^T = B^T A^T, \quad (A^T)^T = A$$
Note the reversal of order in the product rule for transpose: the transpose of a product is the product of the transposes in reverse order.
A square matrix A is symmetric if $A^T = A$, so that $a_{ij} = a_{ji}$ for all i, j. A square matrix A is skew-symmetric if $A^T = -A$, so that $a_{ij} = -a_{ji}$ for all i, j.
There are three types of elementary (row or column) operations: - Interchange any two rows (or columns): denoted $R_i \leftrightarrow R_j$. - Multiply all entries of a row (or column) by a non-zero scalar: $R_i \to kR_i$. - Add to one row a scalar multiple of another row: $R_i \to R_i + kR_j$.
These operations are the foundation of the matrix method of solving linear equations and of computing determinants and inverses, developed in later chapters.
| Type | Order | Example | Property |
|---|---|---|---|
| Row matrix | 1 x n | [1 2 3] | One row |
| Column matrix | m x 1 | [1; 2; 3] | One column |
| Square matrix | n x n | [[1, 2], [3, 4]] | m = n |
| Diagonal | n x n | diag(1, 2, 3) | Non-diagonal entries zero |
| Scalar | n x n | 3I | All diagonal entries equal |
| Identity | n x n | I_n | Diagonal entries all 1 |
| Zero | any | O | All entries zero |
| Operation | Condition | Result order |
|---|---|---|
| A + B | same order | same order |
| kA | always | same order |
| AB | columns(A) = rows(B) | rows(A) x columns(B) |
| A^T | always | n x m if A is m x n |
| A symmetric | A square, A^T = A | n x n |
| A skew-symmetric | A square, A^T = -A | n x n, zero diagonal |
| Property | Formula |
|---|---|
| Addition commutative | A + B = B + A |
| Multiplication associative | A(BC) = (AB)C |
| Distribution | A(B + C) = AB + AC |
| Not commutative | AB not equal to BA in general |
| Transpose of product | (AB)^T = B^T A^T |
| Identity | AI = IA = A |
graph TD
A["Matrices"] --> B["Notation and Order"]
A --> C["Types"]
A --> D["Operations"]
A --> E["Transpose"]
A --> F["Symmetric and Skew-Symmetric"]
A --> G["Elementary Operations"]
B --> B1["A = [aij] of order m x n"]
C --> C1["Row, Column, Square"]
C --> C2["Diagonal, Scalar, Identity, Zero"]
D --> D1["Addition: same order"]
D --> D2["Scalar multiplication"]
D --> D3["Multiplication: columns(A) = rows(B)"]
D --> D4["AB not equal to BA"]
E --> E1["(A^T)ij = aji"]
E --> E2["(AB)^T = B^T A^T"]
F --> F1["Symmetric: A^T = A"]
F --> F2["Skew-symmetric: A^T = -A, zero diagonal"]
G --> G1["R_i <-> R_j"]
G --> G2["R_i -> kR_i"]
G --> G3["R_i -> R_i + kR_j"]
Matrices are one of the most widely applicable structures in mathematics because they encode tabular data and linear transformations in a single notational framework. This chapter's core content, the order of a matrix, its types, the operations of addition, scalar multiplication, and especially matrix multiplication, and the transpose with its reversal rule for products, provides the working vocabulary for the rest of the algebra sequence. The classification of square matrices as symmetric and skew-symmetric, and the decomposition of any square matrix into these two parts, is a result of real significance, as is the family of elementary operations that will power the computation of determinants and inverses. Because matrices unify many seemingly unrelated ideas, a thorough grasp of this chapter converts matrix methods into a favourite tool for the board examination and beyond.