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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.

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:

At =

71 52 43 48 89 63

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:

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

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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