# Matix, Matrices

A **matrix** is an ordered set of numbers listed rectangular(square) form.

**Example:** rectangular matrix A 2 x 3

A = |
71 52 43 48 89 63 |

2 x 3 is the dimension of A

A = (a_{ij})_{nxm} - standard notation for a matrix(a_{ij} are the elements of the matrix),
where 0 ≤ i ≤ n, 0 ≤ j ≤ m.

If the rows of a matrix are equal the columns the matrix is called
**square matrix**.

If A is a sqare matrix, the elements a_{11}, a_{22},..., a_{nn} are called **main diagonal** of the matrix,
and a_{1n}, a_{2,n-2},..., a_{n1} are the **second diagonal**.

We say that one matrix is **identity matrix** if the matrix is a sqare matrix and the elements from the main
diagonal are 1 and all other elements are 0. We sign identity matrices with E_{n} or with E.

When all the elements of a matrix are 0, we say that the matrix is **0-matrix** and write 0 for such a matrix.

The matrix A^{t}_{nxm} is the **transpose** of
A_{nxm} if we change all rows of A with
their corresponding columns.
For example the transpose matrix of the A is:

$A^t= \left( \begin{array}{cc}
71 & 48 \\
52 & 89 \\
43 & 63 \end{array} \right)$

An animated example:

A matrix is called **symmetric matrix** if it is square and is equal to its transpose.

If we have two matrix A, and B that all elements of B are equal ot (-1).a_{i,j}

B is called **opposite matrix** of A and we write B = -A

A = (a_{ij})_{mxn}, B = (b_{ij})_{mxn}
the sum of A and B is the matrix:

_{ij}+ b

_{ij})

_{mxn}

if λ is a number, scalar product of λA is the matrix:

_{ij})

_{mxn}

#### Examples

Let A = |
1 2 4 8 |
B = |
11 14 10 15 |

A + B = |
1+11 2+14 4+10 8+15 |
= |
12 16 14 23 |

**Note**: if A and B have
different dimensions we can not write A + B

if λ = 10

A = |
1 2 4 8 |
λA = |
10 20 40 80 |

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