Introduction to Matrices

In previous posts we have discussed about linear equations, Gaussian elimination and how we can using this elimination reduce the matrix into row-echelon form and hence write all the solutions to the system. Now it’s time to consider matrices for their own sake, although some of the motivation comes from linear equations.

Definition  - A rectangular array of number is called a matrix (the plural is matrices), and the numbers are called the entries of the matrix. They are always denoted by the uppercase letters: A,B,C …

For example: 

$$A=\left[ \begin{matrix} 1 & 2 & -1 \\ 0 & 5 & 6 \\ \end{matrix} \right],B=\left[ \begin{matrix} 1 & -1 \\ 0 & 2 \\ \end{matrix} \right],C=\left[ \begin{matrix} 1 \\ 3 \\ 2 \\ \end{matrix} \right]$$

are matrices.

Clearly matrices come in various shapes depending on the number of rows and columns. For example, the matrix A has 4 rows and 4 columns. In general matrix with m rows in n columns is referred to as an m x n matrix or as having size m x n. Thus matrices A,B and C above have sizes 4x4, 3x3, 3x2 respectevly. A matrix of size 1 x n is called a ROW MATRIX, wherehas one of size n x 1 is called a columns matrix. Each entry of a matrix is defined by the row and column in which it lies. The rows are numbered from the top down, and the columns are numbered from left to right. Then the (i,j)-entry of a matrix is the number lying simultaneously in row I and column j. For example:

The (1,2) entry of

$$\left[ \begin{matrix} 1 & -1 \\ 0 & 1 \\ \end{matrix} \right]$$ ,

is -1.

The (2,3) entry of
$$\left[ \begin{matrix} 1 & 2 & -1 \\ 0 & 5 & 6 \\ \end{matrix} \right]$$

is 6.

A special notation has been devised for the entries of a matrix. If A is m x n matrix, and if the (i,j)-entry of A is denoted as aij then A is displayed as follows:

$$A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & \cdots & {{a}_{1n}} \\ {{a}_{21}} & {{a}_{22}} & \cdots & {{a}_{2n}} \\ \vdots & \vdots & {} & \vdots \\ {{a}_{m1}} & {{a}_{m2}} & \cdots & {{a}_{mn}} \\ \end{matrix} \right].$$

This is usually denoted simply as A =[aij]. Thus aij is the entry in row i and column j of A. For example, 3 x 4 matrix in this notation is written.
 $$A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} & {{a}_{14}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} & {{a}_{24}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} & {{a}_{34}} \\ \end{matrix} \right]$$

According to convention the rows of matrix are mentioned before columns. For example: If a matrix has size m x n, it has m rows and n columns. If we speak of the (i,j)-entry of a matrix, it lies in row I and column j. If an entry is denoted aij, the first subscript I refers to the row and the second subscript j to the column in which aij lies. Two matrices A and B are called equal if and only if: 1)      They have the same size 2)      Corresponding entries are equal

If the entries of A and B are written in the form, A=[aij], B=[bij], described earlier then the second condition takes the following form:

 $$\left[ {{a}_{ij}} \right]=\left[ {{b}_{ij}} \right],$$ 

That means,

$${{a}_{ij}}={{b}_{ij}}$$

 for all i and j.