Concept of Matrices

A matrix is a rectangular arrangement of numbers or symbols written in rows and columns. It is used to organise information in a compact form and is widely applied in mathematics, science, economics, computer science, and data handling.

Matrices are especially useful when data has to be arranged systematically or when several quantities are handled together.

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”

Identify an Element

Let

• Order of matrix \[B = 3 \times 2\].
• The element in the 2nd row and 1st column is 9, so \[b_{21} = 9\].
• The element in the 3rd row and 2nd column is 12, so \[b_{32} = 12\].

Construct a 3 × 2 matrix whose elements are given by \[a_{ij}=\frac{1}{2}|i-3j|\].

Solution: In general, a 3 × 2 matrix is given by \[\mathbf{A}= \begin{bmatrix} {a_{11}} & {a_{12}} \\ {a_{21}} & {a_{22}} \\ {a_{31}} & {a_{32}} \end{bmatrix}\]

Now
\[a_{ij}=\frac{1}{2}\mid i-3j\mid\], i = 1, 2, 3 and j = 1, 2.

Therefore

\[a_{11}=\frac{1}{2}\mid1-3\times1\mid=1\]
\[a_{12}=\frac{1}{2}\mid1-3\times2\mid=\frac{5}{2}\]

\[a_{21}=\frac{1}{2}|2-3\times1|=\frac{1}{2}\]
\[a_{22}=\frac{1}{2}\mid2-3\times2\mid=2\]

\[a_{31}=\frac{1}{2}\mid3-3\times1\mid=0\]
\[a_{32}=\frac{1}{2}\mid3-3\times2\mid=\frac{3}{2}\]

Hence, the required matrix is given by \[\mathrm{A}= \begin{bmatrix} 1 & \frac{5}{2} \\ \frac{1}{2} & 2 \\ 0 & \frac{3}{2} \end{bmatrix}\]

• 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.

• a_ij​: Element in the i-th row and j-th column.