Matrices — Definition, Order and Types

In business, data such as sales figures across branches, costs of raw materials, or production quantities across time periods is naturally organised into rows and columns. A matrix is exactly this: a rectangular array of numbers (called elements or entries) arranged in rows and columns, enclosed in brackets.

For example, the sales (in units) of three products across two branches can be written as

Here Branch 1 forms the first row and Branch 2 the second row, while each column stands for one product.

Order of a Matrix

If a matrix has $m$ rows and $n$ columns, we say it is of order $m \times n$ (read "$m$ by $n$"). The matrix $A$ above has order $2 \times 3$. An element in the $i$-th row and $j$-th column is written $a_{ij}$.

Types of Matrices

• Row matrix — exactly one row, order $1 \times n$.
• Column matrix — exactly one column, order $m \times 1$.
• Square matrix — number of rows equals number of columns ($m=n$).
• Diagonal matrix — a square matrix in which every element off the leading diagonal is zero (diagonal elements may be anything, including zero).
• Scalar matrix — a diagonal matrix in which every diagonal element is the same number $k$.
• Unit (identity) matrix, denoted $I$ — a scalar matrix whose diagonal elements are all $1$. For example $I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$.
• Null (zero) matrix, denoted $O$ — every element is $0$, of any order.

Equality of Matrices

Two matrices are equal only when (a) they have the same order, and (b) every corresponding element is equal: $A=B$ means $a_{ij}=b_{ij}$ for every $i,j$. This gives a practical way to find unknowns hidden inside a matrix equation — matching entry by entry produces ordinary algebraic equations.

Recognising order and type quickly is essential before any arithmetic on matrices, because addition and multiplication are only defined when the orders satisfy specific conditions, covered next. Tamil Nadu's Business Mathematics syllabus draws on the same matrix-algebra principles taught nationally in commerce mathematics, so this foundation carries directly into every later use of matrices for solving real business problems.