Matrix

Matrices is a plural form of a matrix, which is a rectangular array or a table where numbers or elements are arranged in rows and columns. They can have any number of columns and rows. Different operations can be performed on matrices such as addition, scalar multiplication, multiplication, transposition, etc.

Definition

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters. The following are some examples of matrices:

$A={\begin{bmatrix}0&5\\-1&4\\2&{\sqrt {2}}\end{bmatrix}}$ $B={\begin{bmatrix}2+i&4&5\\{\frac {2}{3}}&7&8\\{\sqrt {7}}&-{\frac {2}{9}}&-5\end{bmatrix}}$ $C={\begin{bmatrix}x+1&x^{2}&4\\cos\ x&sin\ x+2&tan\ x\end{bmatrix}}$

In the above examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix. Thus $A$ has 3 rows and 2 columns, $B$ has 3 rows and 3 columns while $C$ has 2 rows and 3 columns.

Order of a matrix

A matrix having $m$ rows and $n$ columns is called a matrix of order $m\times n$ or simply $m\times n$ matrix (read as an $m$ by $n$ matrix). With reference to the So above examples of matrices, we have $A$ as $3\times 2$ matrix, $B$ as $3\times 3$ matrix and $C$ as $2\times 3$ matrix. We find that $A$ has $3\times 2=6$ elements, $B$ and $C$ have $3\times 3=9$ and $2\times 3=6$ elements, respectively.

In general, an $m\times n$ matrix has the following rectangular array:

$A={\begin{bmatrix}a_{11}&a_{12}&.....&a_{1n}\\a_{21}&a_{22}&.....&a_{2n}\\a_{31}&a_{32}&.....&a_{3n}\\.&.&.....&.\\.&.&.....&.\\a_{m1}&a_{m2}&.....&a_{mn}\end{bmatrix}}_{m\times n}$

If a matrix has $m$ rows and $n$ columns, then it will have $m\times n$ elements. A matrix is represented by the uppercase letter, in this case, $A$, and the elements in the matrix are represented by the lower case letter and two subscripts representing the position of the element in the number of row and column in the same order, in this case, $a_{ij}$ , where $i$ is the number of rows, and $j$ is the number of columns. For example, in the given matrix $A$, element in the 3rd row and 2nd column would be $a_{32}$, can be verified in the matrix given above.