# Matrices

The use of tables in everyday life is essential. They appear at all times, without the use of tables, would be restricted to a smaller number of information. In Mathematics, the study of tables is given a special name: it is the study of **matrices** .

An array *m* X *n* is a table composed of m rows in columns containing *m* • *n* elements.

## Representation of an array

An array is represented by upper case letters of the Latin alphabet accompanied by its order, that is, the number of rows and columns that it has.

Examples

• A _{3 × 2 } – Matrix represented by the letter A that has 3 rows and 2 columns.

• B _{4 × 6} – Matrix represented by the letter B that has 4 rows and 6 columns.

Each element of an array is represented by the same letter used in the representation of the matrix, but in lowercase letters accompanied by its index, which is indicated by the position it occupies in the matrix. First, mention the line and then the column.

Examples:

• a _{31 } – Element of matrix A that is located in the third row and in the first column.

• b _{15 } – Element of matrix B that is located in the first row and in the fifth column.

In general, the element of a matrix is represented as _{ij} , located on the ith line and on the jth column.

An A _{4 × 5} matrix can still be represented as follows:

Generally, matrix *A* is represented by A = (a _{ij} ) _{m × n} , where 1 ≤ i ≤ me 1 ≤ j ≤ n, with i, j ∈ N, or:

## Special Matrices

### Null matrix

An array with all its elements equal to zero is called **a null array** .

Examples:

A =is a null matrix of order 2 × 5 and can be indicated by A _{2 × 5} .

B =is a null matrix of order 4 × 3 and can be indicated by B _{4 × 3 } .

### Matrix line

The array that has only one row is called a row **array** .

Examples:

C = [1 5 8 -2] is a row array of order 1 × 4.

D = [-3 2] is an array line of order 1 × 2.

### Column matrix

Already the array that has only one column is called a column **array** .

Examples:

A =is a column matrix of order 3 × 1.

R =is a column matrix of order 5 × 1.

### Square matrix

An array is said to be square when it has the same number of rows and columns.

Examples:

is a square matrix 2 × 2, or even matrix of order 2. In this way, we can represent it as A _{2} .

is a 4 × 4 square matrix, or even an array of order 4.

It can also be represented by B _{4} .

In every square matrix, elements whose line and column position are equal, that is, *i = j* , form the so-called **main diagonal** . The other diagonal, in which the elements satisfy the condition *i + j = n + 1* , is called the **secondary diagonal** .

In matrix A _{3} below, we indicate the main (or primary) and secondary diagonals.

### Identity matrix

The square matrix in which all the elements of the main diagonal are unitary, that is, equal to 1, and the others are null, that is, equal to zero, is called **the identity matrix** . Its representation is always given by I _{n} , where *n is* the order of the matrix.

Examples:

is an identity matrix of order 3 and can be represented by I _{3} .

is an identity matrix of order 4 and can be represented by I _{4} .

### Transposed matrix

From a matrix A, it is possible to obtain another matrix, by orderly inverting the rows through the columns. The resulting array is called **the transposed array** .

From a matrix A = (a _{ij} ) _{m × n} , may define the transpose of the matrix as the matrix A ^{t} = (A ^{T }_{ij} ) _{nxm} , and the ^{T }_{ij =} a _{ji} , ie lines of the matrix A ^{T } are neatly equal to the columns of matrix A and the columns of matrix A ^{T } are neatly equal to the lines of matrix A.

Examples:

## Equality between matrices

Two arrays A and B will be equal if their elements of the same position are equal, so that the two arrays have the same order. An equality between matrices is presented as A = B.

Examples:

a) Given the matrices , for what values of *a, b, x* and *y* these matrices will be the same?

Note that the elements of the same index of both matrices should be equal, then:

b) Given the matricesfor which values of *a, b, c, d, x, y, z* and *k* are these matrices the same?