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 a₁₁, a₂₂,..., 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.

A + 0 = A

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:
Transpose of a matrix

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) = 0

A = (a_ij)_mxn, B = (b_ij)_mxn the sum of A and B is the matrix:

A + B = (a_ij + b_ij)_mxn

A + B = B + A

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

λA = (λa_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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