Matrices

Notes

Matrices – Complete Notes

A matrix is one of the most important concepts in algebra. It is used to arrange numbers or mathematical quantities in rows and columns. Matrices are widely used for solving systems of equations and in many branches of mathematics, science, engineering and computer science.

1. Matrix

A matrix is a rectangular arrangement of numbers, symbols or expressions in rows and columns.

A matrix is generally represented by capital letters such as A, B, C, X, Y, etc.

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

It can also be written as:

\[ A=[a_{ij}]_{m\times n} \]

Here:

• m = number of rows
• n = number of columns
• aij = element situated in the i-th row and j-th column

Example

\[ A= \begin{bmatrix} 2 & 4 & 6\\ 1 & 3 & 5 \end{bmatrix} \]

In this matrix:

• Number of rows = 2
• Number of columns = 3

Also:

\[ a_{11}=2,\qquad a_{12}=4,\qquad a_{13}=6 \]\[ a_{21}=1,\qquad a_{22}=3,\qquad a_{23}=5 \]

2. Order of a Matrix

If a matrix contains m rows and n columns, then the order of the matrix is:

\[ \boxed{m\times n} \]

Diagram – Understanding the Order

\[ \underbrace{ \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix} }_{\text{3 columns}} \quad \left. \begin{array}{c} \\[-12pt] \\ \end{array} \right\} \text{2 rows} \]

Therefore:

\[ \boxed{\text{Order}=2\times3} \]

Number of Elements

If the order of a matrix is \(m\times n\), then:

\[ \boxed{\text{Number of elements}=mn} \]

Example: A matrix of order \(3\times4\) contains:

\[ 3\times4=12 \]

3. Construction of a Matrix

Sometimes the elements of a matrix are defined using a formula involving the row number i and column number j.

Example

Construct a \(2\times3\) matrix whose elements are given by:

\[ a_{ij}=2i+j \]

Calculate each element:

\[…