matrices

A matrix is an array of numbers or in other words rectangular arrangement of numbers in the form of horizontal and vertical lines.

Horizontal lines are called rows and vertical lines are called columns. Each number in a matrix is called an element or entry of the matrix. The elements of the matrix are enclosed in bracket [ ]

If a matrix contains m rows and n columns, then we say it is the matrix of order m × n, having a total number of elements = mn.

A matrix of order m × n can be written as \(A = [a_{ij}]_{m\times n}\) , here 'a' represent an element

For example:

\( \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}\)

Is a matrix of order 2 × 2. It has 4 elements. (1,1)^th element is 1,(1,2)^th element is 2, (2,1)^th element is 3 and (2,2)^th element is 4.  Or we can also express elements as:

\(a_{\textrm{11}} = 1, a_{\textrm{12}}=2, a_{\textrm{21}}=3,a_{\textrm{22}}=4\)

Matrix, where the number of rows equals the number of columns, is called a square matrix. 

Example 2:

\(A = \begin{bmatrix} 1 & 2 & 6 & 8 \\\end{bmatrix}\) is a matrix of order 1 × 4

A matrix having only one row is called a row matrix

Example 3:

\( A = \begin{bmatrix} 1 \\ 3 \\ 4 \\ \end{bmatrix}\) is a matrix of order 3 × 1

A matrix having only one column is called a column matrix

Matrix whose each element is zero is called a null matrix. 

A square matrix in which each diagonal element is 1 and all other elements are zero is called a unit matrix:

\( \begin{bmatrix} 1 & 0 \\0 & 1 \\ \end{bmatrix}\)

Two matrices are said to be the same or equal if -

a) Number of rows and columns are equal

b) Corresponding elements are equal, i.e. entries of both the matrix in the same position are equal.

\(A = \begin{bmatrix} 7 & 4 \\ 9 & 4 \\ \end{bmatrix}\) and \(B = \begin{bmatrix} 7 & 4 \\ 9 & 4 \\ \end{bmatrix}\) Here matrices A and B are equal