Business Mathematics and Statistics · Ch 2 — Matrices

Meaning, Order and Notation of a Matrix

This CHSE Odisha Class 11 Commerce Business Mathematics and Statistics chapter introduces matrices — a compact, rectangular way of arranging numbers that businesses use constantly: recording sales of several products across branches, showing costs of raw materials across suppliers, or presenting a firm's assets and liabilities side by side. The Odisha +2 Commerce syllabus for Business Mathematics and Statistics draws on the same mathematical principles found in matrix algebra taught across Indian state-board commerce curricula, though every example and explanation here is written fresh for this chapter.

What is a matrix?

A matrix is a rectangular array of numbers (or algebraic expressions) arranged in rows (horizontal lines) and columns (vertical lines), enclosed within brackets. For instance, a shop tracking the number of Notebooks and Pens sold on two consecutive days could record its data as

A = \begin{pmatrix} 50 & 80 \\ 40 & 60 \end{pmatrix}

where row 1 is Day 1's sales and row 2 is Day 2's sales; column 1 is Notebooks and column 2 is Pens. Matrices are usually named with capital letters ( $A$ , $B$ , $X$ , ...), and the numbers inside are called its elements or entries.

Order of a matrix

If a matrix has $m$ rows and $n$ columns, it is said to be of order $m \times n$ (read " $m$ by $n$ ") — rows are always named first, columns second. The matrix $A$ above has 2 rows and 2 columns, so its order is $2\times2$ . A matrix such as $B=\begin{pmatrix}5&-2&3\\0&4&-1\end{pmatrix}$ has 2 rows and 3 columns, so its order is $2\times3$ .

Naming an element

Each element of a matrix is written $a_{ij}$ , where $i$ is the row number and $j$ is the column number of that element — row before column, always. In matrix $B$ above, $a_{12}=-2$ (row 1, column 2), $a_{23}=-1$ (row 2, column 3), and $a_{21}=0$ (row 2, column 1).

Note

Row first, always

Both the order of a matrix ( $m\times n$ ) and the subscript of an element ( $a_{ij}$ ) always name the ROW before the COLUMN. Reversing this is the single most common notational slip a beginner makes.

A general $m\times n$ matrix is written

A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}

This chapter works with matrices of small order (mostly $2\times2$ and $3\times3$ ) so that every operation can be carried out by hand, but the definitions apply to a matrix of any order.

Definition 1Matrix

A rectangular array of numbers arranged in rows and columns, enclosed in brackets.

Definition 2Order of a Matrix

If a matrix has m rows and n columns, its order is written m × n — rows named first, columns second.

Definition 3Element a_{ij}

The entry in row i and column j of a matrix; the row subscript is always written first.

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