Matrix concept and types of matrices

Matrix concept

Whenever we place an element in rows and columns we use a matricial structure.

For example, any show in which the tickets are numbered makes use of such structures. The stalls are divided in rows and columns. If on our ticket we see Row $23$ , Seat $12$ , it is indicating that our seat is in the row $23$ and column $12$ .

Any table of those that we use in the text editor is nothing but a matrix since it is organized in rows and columns.

Example

The next table has $3$ rows and $4$ columns. The number placed in row $2$ and column $4$ is zero:

$2$	$1$	$5$	$8$
$3$	$2$	$2$	$0$
$2$	$1$	$6$	$4$

So, if we want a table to become a matrix that is representative of some mathematical object, we must put a numerical value in each cell, remove the grid and enclose it all between two big brackets:

$(\begin{array}{cccc} 2 & 1 & 5 & 8 \\ 3 & 2 & 2 & 0 \\ 2 & 1 & 6 & 4 \end{array})$

And we already have one of those matrices usually used in mathematics.

Types of matrices

A matrix with the same number of rows and columns is called a square matrix. For example:

$(\begin{array}{ccc} 3 & 1 & 4 \\ 4 & 0 & - 5 \\ 12 & 23 & 8 \end{array})$ is square,

and a matrix like:

$(\begin{array}{ccc} 3 & 2 & - 1 \\ 1 & 2 & 0 \end{array})$ is not square.

The first one has three rows and three columns, so it is a $3 \times 3$ matrix. It is read “three by three”. To refer to the second matrix, which has two rows and three columns, we say it is a $2 \times 3$ matrix, or a "two by three" matrix.

So, in general, when we talk about an $m \times n$ matrix we are talking about a matrix that has $m$ rows and $n$ columns. This way of calling the matrix is only a convention and it might change from one author to another.

According to this, a matrix $m \times n$ will be square when $m = n$ .

Notation

The matrix is usually named with capital letters:

$A = (\begin{array}{ccc} 3 & 1 & 4 \\ 4 & 0 & - 5 \\ 12 & 2 & 8 \end{array})$

$B = (\begin{array}{ccc} 3 & 2 & - 1 \\ 1 & 2 & 0 \end{array})$

Letters are also used to refer to the elements or entries that form the matrix:

$(\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array})$

It is to be understood that when we work with a big matrix, for example $100 \times 200$ , that is, with $100$ rows and $200$ columns (or more), the use of letters of the alphabet is not practical. For that reason we can use a notation of type $a_{i j}$ in which $i$ represents the row and $j$ represents the column where the element is.

Example

For instance, in the matrix:

$(\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{array}) = (\begin{array}{ccc} 2 & - 1 & 0 \\ 3 & 1 & 8 \end{array})$

the element $a_{22} = 1$ and the element $a_{13} = 0$ .

Example

Verify the value of the following elements in the matrix:

$a_{31} = 3, a_{25} = 4, a_{27} = - 1, a_{45} = 8$

$(\begin{array}{ccccccc} 2 & 4 & 1 & 8 & 5 & 3 & 8 \\ 3 & 6 & 8 & 2 & 4 & 0 & - 1 \\ 3 & 5 & 7 & 1 & - 8 & 0 & 3 \\ 2 & 5 & 7 & 3 & 8 & 1 & 8 \end{array})$

The zero matrix is defined as the one with all its elements being $0$ , regardless of the number of rows and columns that it has.

Two matrices are said to be equal when all their elements are equal.

Example

$(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array})$

is a null matrix or zero matrix.

Example

$(\begin{array}{cc} 1 & 3 \\ 2 & 6 \end{array}) = (\begin{array}{cc} 1 & 3 \\ 2 & 6 \end{array})$

are equal matrix.