Algebraic Formulas

Matrix. Determinants

Definition. A matrix is a rectangular table of numbers.


E.g.
. We can define a matrix thus:

Let M={1, 2, 3, ..., m} and N={1, 2, 3, ..., n}. A: M x N -> R, A(i,j) = ai,j is called matrix type (m, n), with m rows and n columns.

A matrix that has a size equal to one is called vector. A matrix A[1,n] (1 row, n columns) is called line vector and matrix B[m,1] ( one column, m rows) is called column vector.

Examples:

  It is a matrix type 4x3. The item A[3,1] or a3,1 is 12.
is a matrix type  (1, 7)  or row matrix.

A matrix A(m,n) which has m = n  is called square matrix. So a square matrix is the matrix that has the number of lines equal to the number of columns.

Matrix addition

If A and B are two matrix type m x n, then C = A + B, where ci,j = ai,j + bi,j is the sum (where i<m+1, j<n+1).
Exemple:

Multiplication by a constant

Having the matrix and the constant c, we have B = cA, where  bi,j = cai,j which is the multiplication of matrix A by the constant c.  For instance,

Multiplication

Let the matrix A type m x n and B a matrix type n x p. Then their product is C = AB a matrix type m x p, with
ci,j = ai,1b1,j + ai,2b2,j + ... + ai,nb1,n. For instance,

The properties of matrix multiplication

1. - associativity
2. - neutral element, where In  is the unit matrix is defined as
           
3.
4. - distributivity.

A square matrix A of order n is inversely (or non-singular) if there is a square matrix B of order n, so, to have

  AB = In = BA

In this case, the matrix B is called reverse of the matrix A, and is denoted A-1.

Determinants

Let be the matrix   
                            . It is called the determinant  of the matrix A, the number
                                     

We write                         .

Keywords: algebra, formulas, matrix, matrices, determinants

 

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